Displaying 1-10 of 25 results found.
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31.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 */
Number of integer partitions of n into parts that are alternately equal and unequal.
+10
33
1, 1, 2, 1, 2, 2, 3, 2, 4, 2, 5, 4, 6, 4, 8, 5, 10, 6, 12, 8, 16, 9, 18, 12, 22, 14, 28, 16, 33, 20, 40, 24, 48, 28, 56, 34, 67, 40, 80, 46, 94, 56, 110, 64, 130, 75, 152, 88, 176, 102, 206, 118, 238, 138, 276, 159, 320, 182, 368, 210, 424, 242, 488, 276, 558
COMMENTS
Also partitions whose multiplicities are all 2's, except possibly for the last, which may be 1.
EXAMPLE
The a(1) = 1 through a(12) = 6 partitions (A..C = 10..12):
1 2 3 4 5 6 7 8 9 A B C
11 22 221 33 331 44 441 55 443 66
2211 332 442 551 552
3311 3322 33221 4422
4411 5511
332211
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], And@@Table[#[[i]]==#[[i+1]], {i, 1, Length[#]-1, 2}]&&And@@Table[#[[i]]!=#[[i+1]], {i, 2, Length[#]-1, 2}]&]], {n, 0, 30}]
Number of integer partitions of n into parts that are alternately unequal and equal.
+10
31
1, 1, 1, 2, 3, 4, 4, 6, 7, 9, 10, 12, 14, 16, 18, 22, 25, 28, 31, 36, 40, 46, 50, 56, 64, 71, 78, 88, 96, 106, 118, 130, 143, 158, 172, 190, 209, 228, 248, 274, 298, 324, 354, 384, 418, 458, 494, 536, 584, 631, 683, 742, 800, 864, 936, 1010, 1088, 1176, 1264
EXAMPLE
The a(1) = 1 through a(11) = 12 partitions (A = 10, B = 11):
1 2 3 4 5 6 7 8 9 A B
21 31 32 42 43 53 54 64 65
211 41 51 52 62 63 73 74
311 411 61 71 72 82 83
322 422 81 91 92
511 611 522 433 A1
3221 711 622 533
4221 811 722
32211 5221 911
42211 4331
6221
52211
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], And@@Table[#[[i]]==#[[i+1]], {i, 2, Length[#]-1, 2}]&&And@@Table[#[[i]]!=#[[i+1]], {i, 1, Length[#]-1, 2}]&]], {n, 0, 30}]
CROSSREFS
The alternately equal and unequal version is A351005, even-length A035457. Cf. A000070, A003242, A018819, A027383, A035363, A088218, A122134, A344605, A350842, A350844, A351011.
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))};
Number of integer compositions of n into parts that are alternately equal and unequal.
+10
19
1, 1, 2, 1, 3, 3, 5, 5, 9, 7, 17, 14, 28, 25, 49, 42, 87, 75, 150, 132, 266, 226, 466, 399, 810, 704, 1421, 1223, 2488, 2143, 4352, 3759, 7621, 6564, 13339, 11495, 23339, 20135, 40852, 35215, 71512, 61639, 125148, 107912, 219040, 188839, 383391, 330515, 670998
FORMULA
G.f.: (1 + Sum_{k>0} (x^k)/(1 + x^(2*k)))/(1 - Sum_{k>0} (x^(2*k))/(1 + x^(2*k))). - John Tyler Rascoe, May 28 2024
EXAMPLE
The a(1) = 1 through a(8) = 9 compositions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (22) (113) (33) (115) (44)
(112) (221) (114) (223) (116)
(1122) (331) (224)
(2211) (11221) (332)
(1133)
(3311)
(22112)
(112211)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@Table[#[[i]]==#[[i+1]], {i, 1, Length[#]-1, 2}]&&And@@Table[#[[i]]!=#[[i+1]], {i, 2, Length[#]-1, 2}]&]], {n, 0, 15}]
PROG
(PARI)
C_x(N) = {my(x='x+O('x^N), h=(1+sum(k=1, N, (x^k)/(1+x^(2*k))))/(1-sum(k=1, N, (x^(2*k))/(1+x^(2*k))))); Vec(h)}
CROSSREFS
Without equal relations we have A016116, equal only A001590 (apparently). The version for partitions is A351005. A357621 gives half-alternating sum of standard compositions, skew A357623.
Cf. A029862, A035544, A097805, A122129, A122134, A122135, A351003, A351004, A351007, A357136, A357641.
Number of integer compositions of n into parts that are alternately unequal and equal.
+10
19
1, 1, 1, 3, 4, 7, 8, 13, 17, 25, 30, 44, 58, 77, 98, 142, 176, 245, 311, 426, 548, 758, 952, 1319, 1682, 2308, 2934, 4059, 5132, 7087, 9008, 12395, 15757, 21728, 27552, 38019, 48272, 66515, 84462, 116467, 147812, 203825, 258772, 356686, 452876, 624399, 792578
EXAMPLE
The a(1) = 1 through a(7) = 13 compositions:
(1) (2) (3) (4) (5) (6) (7)
(12) (13) (14) (15) (16)
(21) (31) (23) (24) (25)
(211) (32) (42) (34)
(41) (51) (43)
(122) (411) (52)
(311) (1221) (61)
(2112) (133)
(322)
(511)
(2113)
(3112)
(12211)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@Table[#[[i]]==#[[i+1]], {i, 2, Length[#]-1, 2}]&&And@@Table[#[[i]]!=#[[i+1]], {i, 1, Length[#]-1, 2}]&]], {n, 0, 10}]
CROSSREFS
The version for partitions is A351006. A357621 gives half-alternating sum of standard compositions, skew A357623.
Cf. A001590, A029862, A035544, A097805, A122129, A122134, A122135, A351003, A351004, A351007, A357136, A357641.
Number of integer partitions y of n such that y_i = y_{i+1} for all even i.
+10
18
1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 28, 36, 42, 51, 62, 75, 88, 106, 124, 147, 173, 202, 236, 278, 320, 371, 431, 497, 572, 661, 756, 867, 993, 1132, 1291, 1474, 1672, 1898, 2155, 2439, 2756, 3117, 3512, 3957, 4458, 5008, 5624, 6316, 7072, 7919, 8862, 9899
EXAMPLE
The a(1) = 1 through a(7) = 11 partitions:
(1) (2) (3) (4) (5) (6) (7)
(11) (21) (22) (32) (33) (43)
(111) (31) (41) (42) (52)
(211) (311) (51) (61)
(1111) (2111) (222) (322)
(11111) (411) (511)
(3111) (2221)
(21111) (4111)
(111111) (31111)
(211111)
(1111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], And@@Table[#[[i]]==#[[i+1]], {i, 2, Length[#]-1, 2}]&]], {n, 0, 10}]
CROSSREFS
The ordered version (compositions) is A027383. The version for unequal instead of equal is A122135, even-length A351008. Requiring inequalities at odd positions gives A351006, even-length A351007.
Number of even-length integer partitions of n into parts that are alternately unequal and equal.
+10
17
1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 10, 13, 14, 16, 18, 20, 23, 27, 28, 32, 37, 40, 44, 51, 54, 60, 67, 73, 81, 90, 96, 107, 118, 127, 139, 154, 166, 181, 198, 213, 232, 256, 273, 297, 325, 348, 377, 411, 440, 476, 516, 555, 598, 647, 692, 746, 807
COMMENTS
These are partitions whose multiplicities begin with a 1, are followed by any number of 2's, and end with another 1.
EXAMPLE
The a(3) = 1 through a(15) = 13 partitions (A..E = 10..14):
21 31 32 42 43 53 54 64 65 75 76 86 87
41 51 52 62 63 73 74 84 85 95 96
61 71 72 82 83 93 94 A4 A5
3221 81 91 92 A2 A3 B3 B4
4221 5221 A1 B1 B2 C2 C3
4331 4332 C1 D1 D2
6221 5331 5332 5441 E1
7221 6331 6332 5442
8221 7331 6441
9221 7332
8331
A221
433221
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]], {i, 2, Length[#]-1, 2}]&&And@@Table[#[[i]]!=#[[i+1]], {i, 1, Length[#]-1, 2}]&]], {n, 0, 30}]
CROSSREFS
The alternately equal and unequal version is A035457, any length A351005.
Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.
+10
14
1, 1, 2, 2, 3, 3, 5, 4, 7, 7, 10, 9, 15, 13, 21, 19, 28, 26, 40, 35, 54, 49, 72, 64, 97, 87, 128, 115, 167, 151, 220, 195, 284, 256, 366, 328, 469, 421, 598, 537, 757, 682, 959, 859, 1204, 1085, 1507, 1354, 1880, 1694, 2338, 2104, 2892, 2609, 3574, 3218, 4394
COMMENTS
These are partitions of n with all even multiplicities (or run-lengths), except possibly the last.
EXAMPLE
The a(1) = 1 through a(9) = 7 partitions:
1 2 3 4 5 6 7 8 9
11 111 22 221 33 331 44 333
1111 11111 222 22111 332 441
2211 1111111 2222 22221
111111 3311 33111
221111 2211111
11111111 111111111
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], And@@Table[#[[i]]==#[[i+1]], {i, 1, Length[#]-1, 2}]&]], {n, 0, 30}]
CROSSREFS
The ordered version (compositions) is A016116. The version for unequal instead of equal is A122129, even-length A351008. The version for even instead of odd indices is A351003, even-length A351012.
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