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Coefficients of the '3rd-order' mock theta function psi(q)
+10
67
0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 19, 22, 24, 27, 31, 34, 37, 42, 46, 51, 57, 62, 68, 76, 83, 91, 101, 109, 120, 132, 143, 156, 171, 186, 202, 221, 239, 259, 283, 306, 331, 360, 388, 420, 455, 490, 529, 572, 616, 663, 716, 769, 827
COMMENTS
Number of partitions of n into odd parts such that if a number occurs as a part then so do all smaller positive odd numbers.
Number of ways to express n as a partial sum of 1 + [1,3] + [1,5] + [1,7] + [1,9] + .... E.g., a(6)=2 because we have 6 = 1+1+1+1+1+1 = 1+3+1+1. - Jon Perry, Jan 01 2004 Also number of partitions of n such that the largest part occurs exactly once and all the other parts occur exactly twice. Example: a(9)=4 because we have [9], [7,1,1], [5,2,2] and [3,2,2,1,1]. - Emeric Deutsch, Mar 08 2006 Number of partitions (d1,d2,...,dm) of n such that 0 < d1/1 < d2/2 < ... < dm/m. - Seiichi Manyama, Mar 17 2018
REFERENCES
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.13).
Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31.
FORMULA
G.f.: psi(q) = Sum_{n>=1} q^(n^2) / ( (1-q)*(1-q^3)*...*(1-q^(2*n-1)) ).
G.f.: Sum_{k>=1} q^k*Product_{j=1..k-1} (1+q^(2*j)) (see the Fine reference, p. 58, Eq. (26,53)). - Emeric Deutsch, Mar 08 2006
EXAMPLE
q + q^2 + q^3 + 2*q^4 + 2*q^5 + 2*q^6 + 3*q^7 + 3*q^8 + 4*q^9 + ...
n | Partition (d1,d2,...,dm) | (d1/1, d2/2, ... , dm/m)
--+--------------------------+-------------------------
1 | (1) | (1)
2 | (2) | (2)
3 | (3) | (3)
4 | (4) | (4)
| (1, 3) | (1, 3/2)
5 | (5) | (5)
| (1, 4) | (1, 2)
6 | (6) | (6)
| (1, 5) | (1, 5/2)
7 | (7) | (7)
| (1, 6) | (1, 3)
| (2, 5) | (2, 5/2)
8 | (8) | (8)
| (1, 7) | (1, 7/2)
| (2, 6) | (2, 3)
9 | (9) | (9)
| (1, 8) | (1, 4)
| (2, 7) | (2, 7/2)
| (1, 3, 5) | (1, 3/2, 5/3) (End)
MAPLE
f:=n->q^(n^2)/mul((1-q^(2*i+1)), i=0..n-1); add(f(i), i=1..6);
# second Maple program:
b:= proc(n, i) option remember; (s-> `if`(n>s, 0, `if`(n=s, 1,
b(n, i-1)+b(n-i, min(n-i, i-1)))))(i*(i+1)/2)
end:
a:= n-> `if`(n=0, 0, add(b(j, min(j, n-2*j-1)), j=0..iquo(n, 2))):
MATHEMATICA
Series[Sum[q^n^2/Product[1-q^(2k-1), {k, 1, n}], {n, 1, 10}], {q, 0, 100}]
(* Second program: *)
b[n_, i_] := b[n, i] = Function[s, If[n > s, 0, If[n == s, 1, b[n, i - 1] + b[n - i, Min[n - i, i - 1]]]]][i*(i + 1)/2];
a[n_] := If[n==0, 0, Sum[b[j, Min[j, n-2*j-1]], {j, 0, Quotient[n, 2]}]];
PROG
(PARI) { n=20; v=vector(n); for (i=1, n, v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2, n, k=length(v[i-1]); for (j=1, k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]+2*i-1)); c=vector(n); for (i=1, n, for (j=1, 2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } \\ Jon Perry(PARI) {a(n) = local(t); if(n<0, 0, t = 1 + O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k-1) / (1 - x^(2*k-1)) + O(x^(n-(k-1)^2+1))), n))} /* Michael Somos, Sep 04 2007 */
Expansion of 1 + Sum_{k>0} x^k^2/((1-x)(1-x^2)...(1-x^(2k))).
+10
26
1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 15, 19, 24, 30, 37, 46, 57, 69, 84, 102, 123, 148, 177, 211, 252, 299, 353, 417, 491, 576, 675, 789, 920, 1071, 1244, 1442, 1670, 1929, 2224, 2562, 2946, 3381, 3876, 4437, 5072, 5791, 6602, 7517, 8551, 9714, 11021, 12493, 14145
COMMENTS
Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
In Watson 1937 page 275 he writes "Psi_0(1,q) = prod_1^oo (1+q^{2n}) G(q^8)" so this is the expansion in powers of q^2. - Michael Somos, Jun 28 2015 This appears to be the number of integer partitions of n with every other pair of adjacent parts strictly decreasing, as in the pattern a > b >= c > d >= e for a partition (a, b, c, d, e). For example, the a(1) = 1 through a(9) = 12 partitions are:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(21) (31) (32) (42) (43) (53) (54)
(211) (41) (51) (52) (62) (63)
(311) (321) (61) (71) (72)
(411) (322) (422) (81)
(421) (431) (432)
(511) (521) (522)
(611) (531)
(3221) (621)
(711)
(4221)
(32211)
The even-length case is A351008. The odd-length case appears to be A122130. Swapping strictly and weakly decreasing relations appears to give A122135. The alternately unequal and equal case is A351006, strict A035457, opposite A351005, even-length A351007. (End)
REFERENCES
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.7). MR0858826 (88b:11063)
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(a), p. 591.
FORMULA
Euler transform of period 20 sequence [ 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, ...].
Expansion of f(-x^2) * f(-x^20) / (f(-x) * f(-x^4,-x^16)) in powers of x where f(,) is the Ramanujan general theta function.
Expansion of f(x^3, x^7) / f(-x, -x^4) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, Jun 28 2015 Expansion of f(-x^8, -x^12) / psi(-x) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jun 28 2015 Expansion of G(x^4) / chi(-x) in powers of x where chi() is a Ramanujan theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jun 28 2015 G.f.: Sum_{k>=0} x^k^2 / ((1 - x) * (1 - x^2) ... (1 - x^(2*k))).
G.f.: 1 / (Product_{k>0} (1 - x^(2*k-1)) * (1 - x^(20*k-4)) * (1 - x^(20*k-16))).
Let f(n) = 1/Product_{k >= 0} (1 - q^(20k+n)). Then g.f. is f(1)*f(3)*f(4)*f(5)*f(7)*f(9)*f(11)*f(13)*f(15)*f(16)*f(17)*f(19). - N. J. A. Sloane, Mar 19 2012 a(n) is the number of partitions of n into parts that are either odd or == +-4 (mod 20). - Michael Somos, Jun 28 2015 a(n) ~ (3+sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015
EXAMPLE
Clark Kimberling's SE partition comment, n=6: the 5 SE partitions are [1,1,1,1,1,1] from the partitions 6 and 1^6; [1,1,1,2,1] from 5,1 and 2,1^4; [1,1,3,1] from 4,2 and 2^2,1^2; [2,3,1] from 3,2,1 and 3^2 and 2^3; and [1,2,2,1] from 4,1^2 and 3,1^3. - Wolfdieter Lang, Mar 20 2014 G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 9*x^8 + ...
G.f. = 1/q + q^39 + q^79 + 2*q^119 + 3*q^159 + 4*q^199 + 5*q^239 + ...
MAPLE
f:=n->1/mul(1-q^(20*k+n), k=0..20);
f(1)*f(3)*f(4)*f(5)*f(7)*f(9)*f(11)*f(13)*f(15)*f(16)*f(17)*f(19);
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*[0, 1, 0,
1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1]
[1+irem(d, 20)], d=divisors(j)) *a(n-j), j=1..n)/n)
end:
MATHEMATICA
a[0] = 1; a[n_] := a[n] = Sum[Sum[d*{0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1}[[1+Mod[d, 20]]], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jan 10 2014, after Alois P. Heinz *) a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / QPochhammer[ x, x, 2 k], {k, 0, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Jun 28 2015 *) a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, x^2] QPochhammer[ x^4, x^20] QPochhammer[ x^16, x^20]), {x, 0, n}]; (* Michael Somos, Jun 28 2015 *)
PROG
(PARI)
{a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n), x^k^2 / prod(i=1, 2*k, 1 - x^i, 1 + x * O(x^(n-k^2)))), n))};
Expansion of f(x, -x^4) / phi(-x^2) in powers of x where f(, ) and phi() are Ramanujan theta functions.
+10
21
1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 20, 26, 31, 40, 48, 60, 72, 89, 106, 130, 154, 186, 220, 264, 310, 370, 433, 512, 598, 704, 818, 958, 1110, 1293, 1494, 1734, 1996, 2308, 2650, 3052, 3496, 4014, 4584, 5248, 5980, 6825, 7760, 8834, 10020, 11380, 12882, 14594
COMMENTS
Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
Conjecture: Also the number of integer partitions y of n such that y_i > y_{i+1} for all even i. For example, the a(1) = 1 through a(9) = 12 partitions are:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(11) (21) (22) (32) (33) (43) (44) (54)
(31) (41) (42) (52) (53) (63)
(221) (51) (61) (62) (72)
(321) (331) (71) (81)
(2211) (421) (332) (432)
(3211) (431) (441)
(521) (531)
(3311) (621)
(4211) (3321)
(4311)
(5211)
The even-length case appears to be A122134. (End)
REFERENCES
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.5). MR0858826 (88b:11063)
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(d), p. 591.
FORMULA
Expansion of f(x^2, x^8) / f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Nov 12 2016 Expansion of f(-x^3, -x^7) * f(-x^4, -x^16) / ( f(-x) * f(-x^20) ) in powers of x where f(, ) is Ramanujan's general theta function.
Euler transform of period 20 sequence [ 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, ...].
G.f.: Sum_{k>=0} x^(k^2 + k) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2*k+1))).
Let f(n) = 1/Product_{k >= 0} (1-q^(20k+n)). Then g.f. is f(1)*f(2)*f(5)*f(6)*f(8)*f(9)*f(11)*f(12)*f(14)*f(15)*f(18)*f(19); - N. J. A. Sloane, Mar 19 2012. a(n) ~ (3 + sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Nov 12 2016
EXAMPLE
G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 7*x^7 + 10*x^8 + ...
G.f. = q^9 + q^49 + 2*q^89 + 2*q^129 + 3*q^169 + 4*q^209 + 6*q^249 + ...
MAPLE
f:=n->1/mul(1-q^(20*k+n), k=0..20);
f(1)*f(2)*f(5)*f(6)*f(8)*f(9)*f(11)*f(12)*f(14)*f(15)*f(18)*f(19);
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, -x^5] QPochhammer[ x^4, -x^5] QPochhammer[-x^5] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* Michael Somos, Nov 12 2016 *) nmax = 50; CoefficientList[Series[Product[1/((1 - x^(20*k+1))*(1 - x^(20*k+2))*(1 - x^(20*k+5))*(1 - x^(20*k+6))*(1 - x^(20*k+8))*(1 - x^(20*k+9))*(1 - x^(20*k+11))*(1 - x^(20*k+12))*(1 - x^(20*k+14))*(1 - x^(20*k+15))*(1 - x^(20*k+18))*(1 - x^(20*k+19)) ), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2016 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n+1) - 1) \2, x^(k^2 + k) / prod(i=1, 2*k+1, 1 - x^i, 1 + x * O(x^(n-k^2-k)))), n))};
Expansion of Sum_{k>=0} x^(k^2+k)/((1-x)(1-x^2)...(1-x^(2k))).
+10
20
1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 16, 18, 24, 28, 36, 42, 54, 62, 78, 91, 112, 130, 159, 184, 222, 258, 308, 356, 424, 488, 576, 664, 778, 894, 1044, 1196, 1389, 1590, 1838, 2098, 2419, 2754, 3162, 3596, 4114, 4668, 5328, 6032, 6864, 7760, 8806, 9936, 11252
COMMENTS
Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
In Watson 1937 page 275 he writes "Psi_0(q^{1/2},q) = prod_1^oo (1+q^{2n}) G(-q^2)" so this is the expansion in powers of q^2. - Michael Somos, Jun 29 2015 Conjecture: Also the number of even-length integer partitions y of n such that y_i != y_{i+1} for all even i. For example, the a(2) = 1 through a(9) = 7 partitions are:
(11) (21) (22) (32) (33) (43) (44) (54)
(31) (41) (42) (52) (53) (63)
(51) (61) (62) (72)
(2211) (3211) (71) (81)
(3311) (3321)
(4211) (4311)
(5211)
This appears to be the even-length version of A122135. (End)
REFERENCES
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(c), p. 591.
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.6). MR0858826 (88b:11063)
FORMULA
Euler transform of period 20 sequence [ 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, ...].
Expansion of f(x^4, x^6) / f(-x^2, -x^3) in powers of x where f(, ) is the Ramanujan general theta function. - Michael Somos, Jun 29 2015 Expansion of f(-x^2, x^3) / phi(-x^2) in powers of x where phi() is a Ramanujan theta function. - Michael Somos, Jun 29 2015 Expansion of G(-x) / chi(-x) in powers of x where chi() is a Ramanujan theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jun 29 2015 G.f.: Sum_{k>=0} x^(k^2 + k) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2*k))).
Expansion of f(-x, -x^9) * f(-x^8, -x^12) / ( f(-x) * f(-x^20) ) in powers of x where f(, ) is the Ramanujan general theta function.
a(n) = number of partitions of n into parts that are each either == 2, 3, ..., 7 (mod 20) or == 13, 14, ..., 18 (mod 20). - Michael Somos, Jun 29 2015 [corrected by Vaclav Kotesovec, Nov 12 2016] a(n) ~ (3 - sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Nov 12 2016
EXAMPLE
G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + 7*x^9 + ...
G.f. = q + q^81 + q^121 + 2*q^161 + 2*q^201 + 4*q^241 + 4*q^281 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2 + k) / QPochhammer[ x, x, 2 k], {k, 0, (Sqrt[ 4 n + 1] - 1) / 2}], {x, 0, n}]]; (* Michael Somos, Jun 29 2015 *) a[ n_] := SeriesCoefficient [ 1 / (QPochhammer[ x^4, -x^5] QPochhammer[ -x, -x^5] QPochhammer[ x, x^2]), {x, 0, n}]; (* Michael Somos, Jun 29 2015 *) a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, -x^5] QPochhammer[ -x^3, -x^5] QPochhammer[ -x^5] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* Michael Somos, Jun 29 2015 *) nmax = 50; CoefficientList[Series[Product[1/((1 - x^(20*k+2))*(1 - x^(20*k+3))*(1 - x^(20*k+4))*(1 - x^(20*k+5))*(1 - x^(20*k+6))*(1 - x^(20*k+7))*(1 - x^(20*k+13))*(1 - x^(20*k+14))*(1 - x^(20*k+15))*(1 - x^(20*k+16))*(1 - x^(20*k+17)) *(1 - x^(20*k+18))), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2016 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n + 1) - 1)\2, x^(k^2 + k) / prod(i=1, 2*k, 1 - x^i, 1 + x * O(x^(n -k^2-k)))), n))};
Alternately strict partitions. Number of even-length integer partitions y of n such that y_i > y_{i+1} for all odd i.
+10
12
1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 19, 23, 28, 34, 41, 50, 60, 71, 85, 102, 120, 142, 168, 197, 231, 271, 316, 369, 429, 497, 577, 668, 770, 888, 1023, 1175, 1348, 1545, 1767, 2020, 2306, 2626, 2990, 3401, 3860, 4379, 4963, 5616, 6350, 7173, 8093
EXAMPLE
The a(3) = 1 through a(13) = 12 partitions (A..C = 10..12):
21 31 32 42 43 53 54 64 65 75 76
41 51 52 62 63 73 74 84 85
61 71 72 82 83 93 94
3221 81 91 92 A2 A3
4221 4321 A1 B1 B2
5221 4331 4332 C1
5321 5331 5332
6221 5421 5431
6321 6331
7221 6421
7321
8221
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&And@@Table[#[[i]]!=#[[i+1]], {i, 1, Length[#]-1, 2}]&]], {n, 0, 30}]
CROSSREFS
The version for equal instead of unequal is A035363. The alternately equal and unequal version is A035457, any length A351005. The odd-length version appears to be A122130. The alternately unequal and equal version is A351007, any length A351006. Cf. A000070, A003242, A018819, A027383, A053251, A122134, A350842, A350844, A351003, A351004, A351012.
Number of palindromic Carlitz compositions of n.
+10
5
1, 1, 1, 1, 2, 3, 2, 5, 5, 7, 10, 14, 14, 25, 26, 42, 48, 75, 79, 132, 142, 226, 252, 399, 432, 704, 760, 1223, 1336, 2143, 2328, 3759, 4079, 6564, 7150, 11495, 12496, 20135, 21874, 35215, 38310, 61639, 67018, 107912, 117298, 188839, 205346, 330515, 359350, 578525, 628951
COMMENTS
A palindromic composition is a composition that is identical to its own reverse. There are 2^floor(n/2) palindromic compositions. A Carlitz composition has no two consecutive equal parts ( A003242). This sequence enumerates compositions that are both palindromic and Carlitz. Also the number of odd-length integer compositions of n into parts that are alternately unequal and equal (n > 0). The unordered version (partitions) is A053251. - Gus Wiseman, Feb 26 2022
REFERENCES
S. Heubach and T. Mansour, Compositions of n with parts in a set, Congr. Numer. 168 (2004), 127-143.
S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2010, page 67.
FORMULA
G.f.: (1 + Sum_{j>=1} x^j*(1-x^j)/(1+x^(2*j))) / (1 - Sum_{j>=1} x^(2*j)/(1+x^(2*j))).
a(n) ~ c / r^n, where r = 0.7558768372943356987836792261127971643747976345582722756032673... is the root of the equation sum_{j>=1} x^(2*j)/(1+x^(2*j)) = 1, c = 0.5262391407444644722747255167331403939384758635340487280277... if n is even and c = 0.64032989654153238794063877354074732669441634551692765196197... if n is odd. - Vaclav Kotesovec, Aug 22 2014
EXAMPLE
a(9) = 7 because we have: 9, 1+7+1, 2+5+2, 4+1+4, 1+3+1+3+1, 2+1+3+1+2, 1+2+3+2+1. 2+3+4 is not counted because it is not palindromic. 3+3+3 is not counted because it has consecutive equal parts.
MAPLE
b:= proc(n, i) option remember; `if`(i=0, 0, `if`(n=0, 1,
add(`if`(i=j, 0, b(n-j, j)), j=1..n)))
end:
a:= n-> `if`(n=0, 1, add(b(i, n-2*i), i=0..n/2)):
MATHEMATICA
nn=50; CoefficientList[Series[(1+Sum[x^j(1-x^j)/(1+x^(2j)), {j, 1, nn}])/(1-Sum[x^(2j)/(1+x^(2j)), {j, 1, nn}]), {x, 0, nn}], x]
(* or *)
Table[Length[Select[Level[Map[Permutations, Partitions[n]], {2}], Apply[And, Table[#[[i]]==#[[Length[#]-i+1]], {i, 1, Floor[Length[#]/2]}]]&&Apply[And, Table[#[[i]]!=#[[i+1]], {i, 1, Length[#]-1}]]&]], {n, 0, 20}]
PROG
(PARI) a(n) = polcoeff((1 + sum(j=1, n, x^j*(1-x^j)/(1+x^(2*j)) + O(x*x^n))) / (1 - sum(j=1, n, x^(2*j)/(1+x^(2*j)) + O(x*x^n))), n); \\ Andrew Howroyd, Oct 12 2017
CROSSREFS
Carlitz compositions are counted by A003242. Palindromic compositions are counted by A016116.
Number of odd-length integer partitions y of n such that y_i > y_{i+1} for all even i.
+10
5
0, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 16, 20, 24, 30, 35, 44, 52, 63, 74, 90, 105, 126, 148, 175, 204, 242, 280, 330, 382, 446, 515, 600, 690, 800, 919, 1060, 1214, 1398, 1595, 1830, 2086, 2384, 2711, 3092, 3506, 3988, 4516, 5122, 5788, 6552, 7388, 8345
EXAMPLE
The a(1) = 1 through a(12) = 10 partitions (A..C = 10..12):
1 2 3 4 5 6 7 8 9 A B C
221 321 331 332 432 442 443 543
421 431 441 532 542 552
521 531 541 551 642
621 631 632 651
721 641 732
731 741
821 831
33221 921
43221
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&And@@Table[#[[i]]>#[[i+1]], {i, 2, Length[#]-1, 2}]&]], {n, 0, 30}]
CROSSREFS
The ordered version (compositions) is A000213 shifted right once. All odd-length partitions are counted by A027193. This appears to be the odd-length case of A122135, even-length A122134. The case that is constant at odd indices:
For equality instead of inequality:
- odd-length: A000009 (except at 0)
Number of odd-length integer partitions of n into parts that are alternately equal and strictly decreasing.
+10
4
0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 3, 5, 4, 6, 4, 8, 6, 9, 6, 12, 7, 14, 10, 16, 11, 20, 13, 24, 16, 28, 18, 34, 21, 40, 26, 46, 30, 56, 34, 64, 41, 75, 48, 88, 54, 102, 64, 118, 73, 138, 84, 159, 98, 182, 112, 210, 128, 242, 148, 276, 168, 318
COMMENTS
Also odd-length partitions whose run-lengths are all 2's, except for the last, which is 1.
EXAMPLE
The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
1 2 3 4 5 6 7 8 9 A B C D E F
221 331 332 441 442 443 552 553 554 663
551 661 662 771
33221 44221 44331
55221
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&And@@Table[If[EvenQ[i], #[[i]]!=#[[i+1]], #[[i]]==#[[i+1]]], {i, Length[#]-1}]&]], {n, 0, 30}]
CROSSREFS
With only equalities we get:
- opposite odd-length: A000009 (except at 0) Without equalities we get:
- opposite any length: A122129 (apparently) - opposite odd-length: A122130 (apparently) - even-length: A122134 (apparently)
Number of odd-length integer partitions y of n that are alternately constant, meaning y_i = y_{i+1} for all odd i.
+10
3
0, 1, 1, 2, 1, 3, 2, 4, 2, 7, 3, 9, 4, 13, 6, 19, 6, 26, 10, 35, 12, 49, 16, 64, 20, 87, 27, 115, 32, 151, 44, 195, 53, 256, 69, 328, 84, 421, 108, 537, 130, 682, 167, 859, 202, 1085, 252, 1354, 305, 1694, 380, 2104, 456, 2609, 564, 3218, 676, 3968, 826, 4863
COMMENTS
These are partitions with all even run-lengths except for the last, which is odd.
EXAMPLE
The a(1) = 1 through a(9) = 7 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(111) (221) (222) (331) (332) (333)
(11111) (22111) (441)
(1111111) (22221)
(33111)
(2211111)
(111111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]], {i, 1, Length[#]-1, 2}]&]], {n, 0, 30}]
CROSSREFS
The ordered version (compositions) is A016116 shifted right once. All odd-length partitions are counted by A027193. Replacing equal with unequal relations appears to give:
The case that is also strict at even indices is:
Expansion of f(-x, -x^4) / psi(-x) where psi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.
+10
1
1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 3, 4, 0, 0, 4, 4, 0, 0, 5, 6, 0, 0, 7, 7, 0, 0, 9, 10, 0, 0, 11, 11, 0, 0, 14, 16, 0, 0, 18, 18, 0, 0, 22, 24, 0, 0, 27, 28, 0, 0, 34, 36, 0, 0, 41, 42, 0, 0, 50, 54, 0, 0, 61, 62, 0, 0, 73
LINKS
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 36, Equ. (4.12). MR0858826 (88b:11063).
FORMULA
Expansion of f(-x^2) * f(-x^5) / ( f(-x^4) * f(-x^2, -x^3) ) in powers of x where f(, ) is Ramanujan's general theta function.
Expansion of ( f(-x^11, -x^19) + x^3 * f(-x, -x^29) ) / f(-x^4) in powers of x where f(, ) is Ramanujan's general theta function.
Euler transform of period 20 sequence [ 0, 0, 1, 0, 0, -1, 1, 1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, 0, ...].
a(4*n + 1) = a(4*n + 2) = 0. a(4*n) = A122134(n). a(4*n + 3) = A122130(n).
EXAMPLE
G.f. = 1 + x^3 + x^7 + x^8 + x^11 + x^12 + 2*x^15 + 2*x^16 + 2*x^19 + 2*x^20 + ...
G.f. = q + q^31 + q^71 + q^81 + q^111 + q^121 + 2*q^151 + 2*q^161 + 2*q^191 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^4] / (QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5]), {x, 0, n}]; (* Michael Somos, Oct 04 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n+1) - 1, x^(k^2 + 2*k) / prod(i=1, k, 1 - x^(4*i), 1 + x * O(x^(n - k^2 - 2*k)))), n))};
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