A261562 -id:A261562 - cp's OEIS frontend

A026007 Expansion of Product_{m>=1} (1 + q^m)^m; number of partitions of n into distinct parts, where n different parts of size n are available.

1, 1, 2, 5, 8, 16, 28, 49, 83, 142, 235, 385, 627, 1004, 1599, 2521, 3940, 6111, 9421, 14409, 21916, 33134, 49808, 74484, 110837, 164132, 241960, 355169, 519158, 755894, 1096411, 1584519, 2281926, 3275276, 4685731, 6682699, 9501979, 13471239, 19044780, 26850921, 37756561, 52955699

Offset: 0

N. J. A. Sloane

In general, for t > 0, if g.f. = Product_{m>=1} (1 + t*q^m)^m then a(n) ~ c^(1/6) * exp(3^(2/3) * c^(1/3) * n^(2/3) / 2) / (3^(2/3) * (t+1)^(1/12) * sqrt(2*Pi) * n^(2/3)), where c = Pi^2*log(t) + log(t)^3 - 6*polylog(3, -1/t). - Vaclav Kotesovec, Jan 04 2016

			For n = 4, we have 8 partitions
  01: [4]
  02: [4']
  03: [4'']
  04: [4''']
  05: [3, 1]
  06: [3', 1]
  07: [3'', 1]
  08: [2, 2']

Alois P. Heinz, 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.
Vaclav Kotesovec, Graph - The asymptotic ratio
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 18.

Cf. A000009, A000027, A026741, A073592, A255528, A261562, A266857, A285223, A304040.
Cf. A000219. - Gary W. Adamson, Jun 13 2009
Cf. A027998, A248882, A248883, A248884.
Cf. A026011, A027346, A027906.
Column k=1 of A284992.

with(numtheory):
b:= proc(n) option remember;
      add((-1)^(n/d+1)*d^2, d=divisors(n))
    end:
a:= proc(n) option remember;
      `if`(n=0, 1, add(b(k)*a(n-k), k=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Aug 03 2013

a[n_] := a[n] = 1/n*Sum[Sum[(-1)^(k/d+1)*d^2, {d, Divisors[k]}]*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Apr 17 2014, after Vladeta Jovovic *)
nmax=50; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*x^k/(k*(1-x^k)^2),{k,1,nmax}]],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

N=66; q='q+O('q^N);
gf= prod(n=1,N, (1+q^n)^n );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

a(n) = (1/n)*Sum_{k=1..n} A078306(k)*a(n-k). - Vladeta Jovovic, Nov 22 2002
G.f.: Product_{m>=1} (1+x^m)^m. Weighout transform of natural numbers (A000027). Euler transform of A026741. - Franklin T. Adams-Watters, Mar 16 2006
a(n) ~ zeta(3)^(1/6) * exp((3/2)^(4/3) * zeta(3)^(1/3) * n^(2/3)) / (2^(3/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where zeta(3) = A002117. - Vaclav Kotesovec, Mar 05 2015

A032302 G.f.: Product_{k>=1} (1 + 2*x^k).

Original entry on oeis.org

1, 2, 2, 6, 6, 10, 18, 22, 30, 42, 66, 78, 110, 138, 186, 254, 318, 402, 522, 654, 822, 1074, 1306, 1638, 2022, 2514, 3058, 3798, 4662, 5658, 6882, 8358, 10062, 12186, 14610, 17534, 21150, 25146, 29994, 35694, 42446, 50178, 59514, 70110, 82758, 97602, 114570, 134262

Offset: 0

Christian G. Bower, Apr 01 1998

nonn

"EFK" (unordered, size, unlabeled) transform of 2,2,2,2,...
Number of partitions into distinct parts of 2 sorts, see example. - Joerg Arndt, May 22 2013
In general, for a fixed integer m > 0, if g.f. = Product_{k>=1} (1 + m*x^k) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m+1)*Pi)*n^(3/4)), where c = Pi^2/6 + log(m)^2/2 + polylog(2, -1/m) = -polylog(2, -m). - Vaclav Kotesovec, Jan 04 2016
Antidiagonal sums of A284593. - Peter Bala, Mar 30 2017

			From _Joerg Arndt_, May 22 2013: (Start)
There are a(7) = 22 partitions of 7 into distinct parts of 2 sorts (format P:S for part:sort):
01:  [ 1:0  2:0  4:0  ]
02:  [ 1:0  2:0  4:1  ]
03:  [ 1:0  2:1  4:0  ]
04:  [ 1:0  2:1  4:1  ]
05:  [ 1:0  6:0  ]
06:  [ 1:0  6:1  ]
07:  [ 1:1  2:0  4:0  ]
08:  [ 1:1  2:0  4:1  ]
09:  [ 1:1  2:1  4:0  ]
10:  [ 1:1  2:1  4:1  ]
11:  [ 1:1  6:0  ]
12:  [ 1:1  6:1  ]
13:  [ 2:0  5:0  ]
14:  [ 2:0  5:1  ]
15:  [ 2:1  5:0  ]
16:  [ 2:1  5:1  ]
17:  [ 3:0  4:0  ]
18:  [ 3:0  4:1  ]
19:  [ 3:1  4:0  ]
20:  [ 3:1  4:1  ]
21:  [ 7:0  ]
22:  [ 7:1  ]
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Moussa Ahmia and Mohammed L. Nadji, Overpartitions into distinct parts, Comb. Number Theory 14 (2025), no. 1, 65-74, preprint (2021).
C. G. Bower, Transforms (2)
Vaclav Kotesovec, Asymptotic formula for A032302
Eric Weisstein's World of Mathematics, Dilogarithm
Eric Weisstein's MathWorld, Polylogarithm
Wikipedia, Polylogarithm

Cf. A000009, A032308, A261562, A261568, A261569, A266576, A284593.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 2*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015
# Alternatively:
simplify(expand(QDifferenceEquations:-QPochhammer(-2,x,99)/3,x)):
seq(coeff(%,x,n), n=0..47); # Peter Luschny, Nov 17 2016

nn=47; CoefficientList[Series[Product[1+2x^i,{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 07 2013 *)
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*2^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(QPochhammer[-2, x]/3 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

N=66; x='x+O('x^N); Vec(prod(n=1,N, 1+2*x^n)) \\ Joerg Arndt, May 22 2013

a(n) = A072706(n)*2 for n>=1.
G.f.: Sum_{n>=0} (2^n*q^(n*(n+1)/2) / Product_{k=1..n} (1-q^k ) ). - Joerg Arndt, Jan 20 2014
a(n) = (1/3) [x^n] QPochhammer(-2,x). - Vladimir Reshetnikov, Nov 20 2015
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Pi^2/6 + log(2)^2/2 + polylog(2, -1/2) = 1.43674636688368094636290202389358335424... . Equivalently, c = A266576 = Pi^2/12 + log(2)^2 + polylog(2, 1/4)/2. - Vaclav Kotesovec, Jan 04 2016

A266891 Expansion of Product_{k>=1} (1 + k*x^k)^k.

Original entry on oeis.org

1, 1, 4, 13, 29, 81, 188, 456, 1030, 2405, 5295, 11611, 25246, 53552, 113332, 235685, 486011, 990840, 2006567, 4018010, 7992003, 15768511, 30875424, 60060509, 116042548, 222817961, 425200270, 806991037, 1522748592, 2858792520, 5339457208, 9924370365

Offset: 0

Vaclav Kotesovec, Jan 05 2016

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n, g(n) = -n. - Seiichi Manyama, Nov 18 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
Vaclav Kotesovec, Graph - The asymptotic ratio (200000 terms)

Cf. A022629, A026007, A032302, A261562, A266964, A304210, A304211.

nmax=50; CoefficientList[Series[Product[(1+k*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
(* More efficient program: *) nmax = 50; s = 1+x; Do[s*=Sum[Binomial[k, j] * k^j * x^(j*k), {j, 0, nmax/k}]; s = Take[Expand[s], Min[nmax + 1, Exponent[s, x] + 1]];, {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Jan 07 2016 *)

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d*(-d)^(1+n/d). - Seiichi Manyama, Nov 18 2017

Conjecture: log(a(n)) ~ n^(2/3) * (2*log(3*n) - 3) / (4*3^(1/3)). - Vaclav Kotesovec, May 08 2018

A071109 Expansion of Product_{k>=1} 1/(1+2*x^k).

Original entry on oeis.org

1, -2, 2, -6, 14, -26, 50, -102, 214, -426, 834, -1678, 3398, -6778, 13482, -27022, 54198, -108306, 216346, -432878, 866334, -1732386, 3463626, -6927926, 13858350, -27715378, 55426002, -110855030, 221719582, -443433610, 886848930, -1773709078, 3547455846

Offset: 0

Sharon Sela (sharonsela(AT)hotmail.com), May 27 2002

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A070933, A032302, A000700, A246935, A261562, A261566, A261582.

nmax = 40; CoefficientList[Series[Product[1/(1 + 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*2^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(O[x]^30 + 3/QPochhammer[-2, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) ~ c * (-2)^n, where c = Product_{j>=1} 1/(1-1/(-2)^j) = 1/QPochhammer[-1/2,-1/2] = 0.8259519860658427384636116224100201356301... . - Vaclav Kotesovec, Aug 25 2015

G.f.: Sum_{i>=0} (-2)^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 13 2018

More terms from Vaclav Kotesovec, Aug 25 2015

A261561 Expansion of Product_{k>=1} (1/(1 - 2*x^k))^k.

Original entry on oeis.org

1, 2, 8, 22, 64, 162, 424, 1022, 2480, 5770, 13336, 30046, 67184, 147554, 321592, 692278, 1479568, 3133474, 6596008, 13788606, 28679264, 59335530, 122256456, 250875550, 513116864, 1046190786, 2127557592, 4316282006, 8739096992, 17661731138, 35639764536

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 1001 terms from Vaclav Kotesovec)

Cf. A000219, A070933, A246935, A261562, A261563, A261565.

b:= proc(n, i) option remember;  `if`(n=0, 1, `if`(i<1, 0,
      add(2^j*binomial(i+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 21 2018

nmax = 50; CoefficientList[Series[Product[(1/(1 - 2*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[2^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

{a(n) = polcoeff( exp( sum(m=1,n,x^m/m * sumdiv(m,d,2^d*m^2/d^2) ) +x*O(x^n)),n)}
for(n=0,40,print1(a(n),", ")) \\ Paul D. Hanna, Sep 30 2015

a(n) ~ c * 2^n, where c = Product_{j>=1} 1/(1 - 1/2^j)^(j+1) = 34.7387234654851595844514193757064296508992247003230539635669599773458896...

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} 2^d * n^2/d^2 ). - Paul D. Hanna, Sep 30 2015

A261563 Expansion of Product_{k>=1} ((1 + 2x^k)/(1 - 2x^k))^k.

Original entry on oeis.org

1, 4, 16, 60, 192, 596, 1744, 4892, 13248, 34868, 89296, 223660, 548928, 1323060, 3137520, 7332332, 16907584, 38517444, 86777328, 193523404, 427562816, 936555044, 2035286576, 4390850268, 9409096576, 20037827876, 42429318480, 89369282460, 187325508288

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A156616, A261561, A261562, A261584.

nmax = 50; CoefficientList[Series[Product[((1 + 2*x^k)/(1 - 2*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[2^(2*k)/(2*k-1)*x^(2*k-1)/(1 - x^(2*k-1))^2, {k, 1, nmax}]], {x, 0, nmax}], x]

{a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, (2^d - (-2)^d) * m^2/d^2) ) +x*O(x^n)), n)}
for(n=0,40,print1(a(n),", ")) \\ Paul D. Hanna, Sep 30 2015

a(n) ~ c * 2^n, where c = 2 * Product_{j>=1} ((1 + 1/2^j)/(1 - 1/2^j))^(j+1) = 1021.5383556752320172813996404366861329314041364322798995039038143325883...

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} (2^d - (-2)^d) * n^2/d^2 ). - Paul D. Hanna, Sep 30 2015

A298987 a(n) = [x^n] Product_{k>=1} (1 + n*x^k)^k.

Original entry on oeis.org

1, 1, 4, 27, 80, 400, 1908, 6223, 31296, 116478, 450100, 1828915, 7360848, 26906828, 95776772, 403908975, 1421758720, 5072014447, 18481180644, 68350964211, 246180936400, 827642046294, 2958748580084, 10294629775620, 36607347335232, 120800714172500, 407951731319860, 1405943613730899

Offset: 0

Ilya Gutkovskiy, Jan 31 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A026007, A261562, A266857, A266891, A291698, A297322, A298985, A298986, A298988.

Table[SeriesCoefficient[Product[(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

A266857 Expansion of Product_{k>=1} (1 + 3*x^k)^k.

Original entry on oeis.org

1, 3, 6, 27, 48, 132, 324, 651, 1491, 3078, 6447, 12795, 25839, 50088, 96099, 184491, 343920, 640545, 1173609, 2138403, 3850584, 6882354, 12186336, 21423660, 37421757, 64816608, 111637392, 190976859, 324868530, 549265290, 923904711, 1545406077, 2572326510

Offset: 0

Vaclav Kotesovec, Jan 04 2016

nonn

In general, for m > 0, if g.f. = Product_{k>=1} (1 + m*x^k)^k then a(n) ~ c^(1/6) * exp(3^(2/3) * c^(1/3) * n^(2/3) / 2) / (3^(2/3) * (m+1)^(1/12) * sqrt(2*Pi) * n^(2/3)), where c = Pi^2*log(m) + log(m)^3 - 6*polylog(3, -1/m).

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A026007, A261562, A261565, A261567.

nmax=50; CoefficientList[Series[Product[(1+3*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c^(1/6) * exp(3^(2/3) * c^(1/3) * n^(2/3) / 2) / (2^(2/3) * 3^(2/3) * sqrt(Pi) * n^(2/3)), where c = Pi^2*log(3) + log(3)^3 - 6*polylog(3, -1/3) = 14.092743327504459346835224018840792668682349056875722467... .

cp's OEIS Frontend

A026007 Expansion of Product_{m>=1} (1 + q^m)^m; number of partitions of n into distinct parts, where n different parts of size n are available.

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A032302 G.f.: Product_{k>=1} (1 + 2*x^k).

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A266891 Expansion of Product_{k>=1} (1 + k*x^k)^k.

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A071109 Expansion of Product_{k>=1} 1/(1+2*x^k).

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A261561 Expansion of Product_{k>=1} (1/(1 - 2*x^k))^k.

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A261563 Expansion of Product_{k>=1} ((1 + 2*x^k)/(1 - 2*x^k))^k.

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A298987 a(n) = [x^n] Product_{k>=1} (1 + n*x^k)^k.

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A266857 Expansion of Product_{k>=1} (1 + 3*x^k)^k.

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A261563 Expansion of Product_{k>=1} ((1 + 2x^k)/(1 - 2x^k))^k.