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Search: a288415 -id:a288415
Displaying 1-9 of 9 results found.
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A321877
Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^sigma_k(j).
+0
2
1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 5, 7, 6, 1, 1, 9, 15, 14, 10, 1, 1, 17, 37, 41, 28, 17, 1, 1, 33, 99, 137, 107, 58, 25, 1, 1, 65, 277, 491, 487, 286, 106, 38, 1, 1, 129, 795, 1829, 2429, 1749, 700, 201, 59, 1, 1, 257, 2317, 6971, 12763, 12056, 5901, 1735, 372, 86
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
FORMULA
G.f. of column k: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^k).
G.f. of column k: exp(Sum_{j>=1} sigma_(k+1)(j)*x^j/(j*(1 - x^(2*j)))).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
2, 3, 5, 9, 17, 33, ...
4, 7, 15, 37, 99, 277, ...
6, 14, 41, 137, 491, 1829, ...
10, 28, 107, 487, 2429, 12763, ...
MATHEMATICA
Table[Function[k, SeriesCoefficient[Product[(1 + x^j)^DivisorSigma[k, j], {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten
Table[Function[k, SeriesCoefficient[Exp[Sum[DivisorSigma[k + 1, j] x^j/(j (1 - x^(2 j))), {j, 1, n}]], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten
CROSSREFS
Columns k=0..9 give A107742, A192065, A288414, A288415, A301548, A301549, A301550, A301551, A301552, A301553.
Main diagonal gives A321042.
Cf. A321876.
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Nov 20 2018
STATUS
approved
A321042
a(n) = [x^n] Product_{k>=1} (1 + x^k)^sigma_n(k).
+0
8
1, 1, 5, 37, 491, 12763, 690756, 70250881, 13805853214, 5567873958982, 4386114219458332, 6711687353310594027, 21048327399504558833175, 131214860796100022696745520, 1603892616451767287785208156624, 40296605442098101265893075903063822, 2031406440758379976992019043333960734724
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..80
FORMULA
a(n) = [x^n] Product_{i>=1, j>=1} (1 + x^(i*j))^(j^n).
a(n) = [x^n] exp(Sum_{k>=1} sigma_(n+1)(k)*x^k/(k*(1 - x^(2*k)))).
MATHEMATICA
Table[SeriesCoefficient[Product[(1 + x^k)^DivisorSigma[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Product[Product[(1 + x^(i j))^(j^n), {j, 1, n}], {i, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n + 1, k] x^k/(k (1 - x^(2 k))), {k, 1, n}]], {x, 0, n}], {n, 0, 16}]
CROSSREFS
Cf. A107742, A192065, A288414, A288415, A301548, A301549, A301550, A301551, A301552, A301553, A319647.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 26 2018
STATUS
approved
A301553
Expansion of Product_{k>=1} (1 + x^k)^(sigma_9(k)).
+0
4
1, 1, 513, 20197, 413669, 12445003, 372981573, 9158438541, 223776496101, 5567873958982, 132009631562091, 3018411978731059, 68171158091244082, 1512439928316217508, 32796174722883608382, 698503712498547606328, 14656105328324700415778, 302787437988353941515934
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..995
FORMULA
a(n) ~ exp(11 * Pi^(10/11) * (31*Zeta(11))^(1/11) * n^(10/11) / (2^(13/11) * 5^(10/11))) * (155*Zeta(11)/Pi)^(1/22) / (2^(155/264) * sqrt(11) * n^(6/11)).
G.f.: exp(Sum_{k>=1} sigma_10(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 26 2018
MATHEMATICA
nmax = 30; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[9, k], {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
Cf. A107742 (m=0), A192065 (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).
Cf. A013957, A301547.
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Mar 23 2018
STATUS
approved
A288423
Expansion of Product_{k>=1} 1/(1 + x^k)^(sigma_3(k)).
+0
4
1, -1, -8, -20, -8, 134, 512, 1062, 406, -5319, -22532, -51843, -58869, 83035, 648412, 1947384, 3665081, 3040131, -8272126, -46481039, -128400098, -234847560, -215189896, 378947363, 2437661943, 7036096665, 13868464378, 16886982518, -4042283985, -93095770772
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..7039
FORMULA
Convolution inverse of A288415.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A288420(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 29 2018
MAPLE
with(numtheory): seq(coeff(series(mul(1/(1+x^k)^(sigma[3](k)), k=1..n), x, n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 31 2018
MATHEMATICA
nmax = 40; CoefficientList[Series[Product[1/(1+x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)
PROG
(PARI) m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1+x^k)^sigma(k, 3))) \\ G. C. Greubel, Oct 30 2018
(Magma) m:=40; R<q>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1+q^k)^DivisorSigma(3, k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
CROSSREFS
Cf. A288415, A288420.
Product_{k>=1} 1/(1 + x^k)^sigma_m(k): A288007 (m=0), A288421 (m=1), A288422 (m=2), this sequence (m=3).
KEYWORD
sign
AUTHOR
Seiichi Manyama, Jun 09 2017
STATUS
approved
A288414
Expansion of Product_{k>=1} (1 + x^k)^(sigma_2(k)).
+0
11
1, 1, 5, 15, 41, 107, 286, 700, 1735, 4162, 9803, 22673, 51822, 116376, 258548, 567197, 1230763, 2642958, 5622616, 11850537, 24769248, 51353095, 105662389, 215838649, 437890022, 882562763, 1767741732, 3519599996, 6967592060, 13717874719, 26865949075
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
FORMULA
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288419(k)*a(n-k) for n > 0.
a(n) ~ exp(2^(5/4) * (7*Zeta(3))^(1/4) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - 5^(1/4) * Pi * n^(1/4) / (2^(13/4) * 3^(7/4) * (7*Zeta(3))^(1/4))) * (7*Zeta(3))^(1/8) / (2^(15/8) * 15^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 23 2018
G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^2). - Ilya Gutkovskiy, Aug 26 2018
MAPLE
with(numtheory): seq(coeff(series(mul((1+x^k)^(sigma[2](k)), k=1..n), x, n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 31 2018
MATHEMATICA
nmax = 40; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[2, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)
PROG
(PARI) m=40; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^sigma(k, 2))) \\ G. C. Greubel, Oct 30 2018
(Magma) m:=40; R<q>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+q^k)^DivisorSigma(2, k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
CROSSREFS
Cf. A275585, A288389, A288419.
Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), A192065 (m=1), this sequence (m=2), A288415 (m=3).
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jun 08 2017
STATUS
approved
A288420
a(n) = Sum_{d|n} d^4*A000593(n/d).
+0
7
1, 17, 85, 273, 631, 1445, 2409, 4369, 6898, 10727, 14653, 23205, 28575, 40953, 53635, 69905, 83539, 117266, 130341, 172263, 204765, 249101, 279865, 371365, 394406, 485775, 558778, 657657, 707311, 911795, 923553, 1118481, 1245505, 1420163, 1520079, 1883154
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Multiplicative because this sequence is the Dirichlet convolution of A000583 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 20 2018
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A027848(n) for odd n.
Multiplicative with a(2^e) = (16^(e+1)-1)/15 and a(p^e) = (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^4*zeta(5)/480 = (3/16)*zeta(4)*zeta(5) = 0.210429... . (End)
MATHEMATICA
f[p_, e_] := (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1); f[2, e_] := (16^(e+1)-1)/15; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)
CROSSREFS
Cf. A000583, A013662, A013663, A027848, A288415.
Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), A288418 (k=2), A288419 (k=3), this sequence (k=4).
KEYWORD
mult,nonn
AUTHOR
Seiichi Manyama, Jun 09 2017
EXTENSIONS
Keyword:mult added by Andrew Howroyd, Jul 23 2018
STATUS
approved
A288391
Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_3(k)).
+0
13
1, 1, 10, 38, 156, 534, 2014, 6796, 23312, 76165, 247234, 780343, 2435903, 7453859, 22538336, 67130594, 197666509, 574876417, 1654464954, 4711217687, 13288453688, 37133349758, 102873771662, 282630567325, 770410193747, 2084205092693, 5598070811010
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
FORMULA
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A027848(k)*a(n-k) for n > 0.
a(n) ~ exp((5*Pi)^(4/5) * Zeta(5)^(1/5) * n^(4/5) / (2^(8/5) * 3^(1/5)) - Zeta'(-3)/2) * Zeta(5)^(121/1200) / ((24*Pi)^(121/1200) * 5^(721/1200) * n^(721/1200)). - Vaclav Kotesovec, Mar 23 2018
G.f.: exp(Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
d*sigma[3](d), d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jun 08 2017
MATHEMATICA
nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 23 2018 *)
PROG
(PARI) m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^sigma(k, 3))) \\ G. C. Greubel, Oct 30 2018
(Magma) m:=40; R<q>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-q^k)^DivisorSigma(3, k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
CROSSREFS
Cf. A027848, A288392, A288415.
Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), A275585 (m=2), this sequence (m=3).
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jun 08 2017
STATUS
approved
A192065
Expansion of Product_{k>=1} Q(x^k)^k where Q(x) = Product_{k>=1} (1 + x^k).
+0
28
1, 1, 3, 7, 14, 28, 58, 106, 201, 372, 669, 1187, 2101, 3624, 6229, 10591, 17796, 29659, 49107, 80492, 131157, 212237, 341084, 544883, 865717, 1367233, 2148552, 3359490, 5227270, 8096544, 12486800, 19174319, 29326306, 44678825, 67811375, 102549673, 154545549
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Euler transform of A002131. - Vaclav Kotesovec, Mar 26 2018
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
FORMULA
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288418(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 09 2017
a(n) ~ exp(3*Pi^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2^(5/3) - Pi^(4/3) * n^(1/3) / (3*2^(7/3) * Zeta(3)^(1/3)) - Pi^2 / (864 * Zeta(3))) * Zeta(3)^(1/6) / (2^(19/24) * sqrt(3) * Pi^(1/6) * n^(2/3)). - Vaclav Kotesovec, Mar 23 2018
MATHEMATICA
nn = 30; b = Table[DivisorSigma[1, n], {n, nn}]; CoefficientList[Series[Product[(1 + x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)
kmax = 37; Product[QPochhammer[-1, x^k]^k/2^k, {k, 1, kmax}] + O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, Jul 03 2017 *)
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[DivisorSum[k, # / GCD[#, 2] &] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)
PROG
(PARI) N=66; x='x+O('x^N);
Q(x)=prod(k=1, N, 1+x^k);
gf=prod(k=1, N, Q(x^k)^k );
Vec(gf) /* Joerg Arndt, Jun 24 2011 */
CROSSREFS
Cf. A061256 (1/Product_{k>=1} P(x^k)^k where P(x) = Product_{k>=1} (1 - x^k)).
Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), this sequence (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).
KEYWORD
nonn
AUTHOR
Joerg Arndt, Jun 24 2011
STATUS
approved
A107742
G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).
+0
57
1, 1, 2, 4, 6, 10, 17, 25, 38, 59, 86, 125, 184, 260, 369, 524, 726, 1005, 1391, 1894, 2576, 3493, 4687, 6272, 8373, 11090, 14647, 19294, 25265, 32991, 42974, 55705, 72025, 92895, 119349, 152965, 195592, 249280, 316991, 402215, 508932, 642598, 809739, 1017850, 1276959, 1599015, 1997943, 2491874, 3102477, 3855165, 4782408, 5922954
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
From Gus Wiseman, Sep 13 2022: (Start)
Also the number of multiset partitions of integer partitions of n into intervals, where an interval is a set of positive integers with all differences of adjacent elements equal to 1. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:
{{1}} {{2}} {{3}} {{4}}
{{1},{1}} {{1,2}} {{1},{3}}
{{1},{2}} {{2},{2}}
{{1},{1},{1}} {{1},{1,2}}
{{1},{1},{2}}
{{1},{1},{1},{1}}
Intervals are counted by A001227, ranked by A073485.
The initial version is A007294.
The strict version is A327731.
The version for gapless multisets instead of intervals is A356941.
The case of strict partitions is A356957.
Also the number of multiset partitions of integer partitions of n into distinct constant blocks. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:
{{1}} {{2}} {{3}} {{4}}
{{1,1}} {{1,1,1}} {{2,2}}
{{1},{2}} {{1},{3}}
{{1},{1,1}} {{1,1,1,1}}
{{2},{1,1}}
{{1},{1,1,1}}
Constant multisets are counted by A000005, ranked by A000961.
The non-strict version is A006171.
The unlabeled version is A089259.
The non-constant block version is A261049.
The version for twice-partitions is A279786, factorizations A296131.
Also the number of multiset partitions of integer partitions of n into constant blocks of odd length. For example, a(1) = 1 through a(4) = 6 multiset partitions are:
{{1}} {{2}} {{3}} {{4}}
{{1},{1}} {{1,1,1}} {{1},{3}}
{{1},{2}} {{2},{2}}
{{1},{1},{1}} {{1},{1,1,1}}
{{1},{1},{2}}
{{1},{1},{1},{1}}
The strict version is A327731 (also).
(End)
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
N. J. A. Sloane, Transforms
FORMULA
Euler transform of A001227.
Weigh transform of A000005.
G.f. satisfies: log(A(x)) = Sum_{n>=1} A109386(n)/n*x^n, where A109386(n) = Sum_{d|n} d*Sum_{m|d} (m mod 2). - Paul D. Hanna, Jun 26 2005
G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1-x^(2n)) /n ). - Paul D. Hanna, Mar 28 2009
G.f.: Product_{n>=1} Q(x^n) where Q(x) is the g.f. of A000009. - Joerg Arndt, Feb 27 2014
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A109386(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 04 2017
Conjecture: log(a(n)) ~ Pi*sqrt(n*log(n)/6). - Vaclav Kotesovec, Aug 29 2018
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1+x^(i*j)), {i, 1, nmax}, {j, 1, nmax/i}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 04 2017 *)
nmax = 50; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 23 2018 *)
nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]]; , {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Aug 28 2018 *)
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Table[Length[Select[Join@@mps/@IntegerPartitions[n], And@@chQ/@#&]], {n, 0, 5}] (* Gus Wiseman, Sep 13 2022 *)
PROG
(PARI) a(n)=polcoeff(prod(k=1, n, prod(j=1, n\k, 1+x^(j*k)+x*O(x^n))), n) /* Paul D. Hanna */
(PARI) N=66; x='x+O('x^N); gf=1/prod(j=0, N, eta(x^(2*j+1))); gf=prod(j=1, N, (1+x^j)^numdiv(j)); Vec(gf) /* Joerg Arndt, May 03 2008 */
(PARI) {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, sigma(m)*x^m/(1-x^(2*m)+x*O(x^n))/m)), n))} /* Paul D. Hanna, Mar 28 2009 */
CROSSREFS
Cf. A006171, A109386, A219554, A280473, A280486, A288007.
Product_{k>=1} (1 + x^k)^sigma_m(k): this sequence (m=0), A192065 (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).
A000041 counts integer partitions, strict A000009.
A000110 counts set partitions.
A072233 counts partitions by sum and length.
Cf. A001055, A001970, A011782, A055887, A061260, A270995, A279784, A294617, A304969.
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Jun 11 2005
EXTENSIONS
More terms from Paul D. Hanna, Jun 26 2005
STATUS
approved
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