A261519 -id:A261519 - cp's OEIS frontend

A156616 G.f.: Product_{n>0} ((1+x^n)/(1-x^n))^n.

1, 2, 6, 16, 38, 88, 196, 420, 878, 1794, 3584, 7032, 13572, 25792, 48352, 89512, 163774, 296444, 531234, 943072, 1659560, 2896376, 5015700, 8622108, 14718652, 24960138, 42062200, 70458160, 117349856, 194381704, 320295312, 525123604

Offset: 0

R. J. Mathar, Feb 11 2009

Generating function for a sum over strict plane partitions weighted with 2 powered to their number of connected components.
The inverse Euler transform is apparently 2, 3, 6, 6, 10, 9, 14, 12, 18, 15, 22, 18, 26, 21, ..., A016825 interlaced with A008585. - R. J. Mathar, Apr 23 2009
In general, for m >= 1, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(m*k), then a(n) ~ exp(m/12 + 3/2 * (7*m*Zeta(3)/2)^(1/3) * n^(2/3)) * m^(1/6 + m/36) * (7*Zeta(3))^(1/6 + m/36) / (A^m * 2^(2/3 + m/9) * sqrt(3*Pi) * n^(2/3 + m/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
In general, for m >= 0, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(k^m), then a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + Zeta'(-m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
Ali H. Al-Saedi, Congruences for restricted plane overpartitions modulo 4 and 8, Raman. J. 48 (2) (2019) 251
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.
Mirjana Vuletic, A generalization of MacMahon's formula, Trans. Am. Math. Soc. 361 (2009) 2789-2804.

Cf. A000219, A015128, A026007, A261384, A261386, A261389.
Cf. A206622, A206623, A206624, A261519, A261520.
Cf. A285462, A285447, A285460, A285461.
Cf. A285446, A285458, A285459.
Cf. A306081.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)

{a(n)=polcoeff(exp(sum(m=1,n,(sigma(2*m,2)-sigma(m,2))/2*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, May 01 2010

Convolve A000219 with A026007.
O.g.f.: exp( Sum_{n>=1} (sigma_2(2n) - sigma_2(n))/2 *x^n/n ), where sigma_2(n) is the sum of squares of divisors of n (A001157). - Paul D. Hanna, May 01 2010
a(n) ~ exp(1/12 + 3 * 2^(-4/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * (7*Zeta(3))^(7/36) / (A * 2^(7/9) * sqrt(3*Pi) * n^(25/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A076577(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 30 2017
G.f.: A(x) = exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 - x^(2*n+1))^2) ). Cf. A000122 and A302237. - Peter Bala, Dec 23 2021

A260916 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(Fibonacci(k)).

Original entry on oeis.org

1, 2, 4, 10, 22, 48, 104, 220, 460, 954, 1956, 3976, 8026, 16084, 32032, 63440, 124974, 245008, 478204, 929452, 1799508, 3471396, 6673724, 12788976, 24433528, 46546738, 88432264, 167575474, 316768948, 597389576, 1124092476, 2110661644, 3955006820, 7396477224

Offset: 0

Vaclav Kotesovec, Aug 18 2015

nonn

Convolution of A166861 and A261050.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015.

Cf. A000045, A001622, A166861, A261050, A261519, A261520.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^Fibonacci[k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ phi^n / (2^(3/4) * 5^(1/8) * sqrt(Pi) * n^(3/4)) * exp(-1/5 + 2*5^(-1/4)*sqrt(2*n) + s), where s = 2 * Sum_{k>=1} phi^(2*k+1) / ((phi^(4*k+2) - phi^(2*k+1) - 1)*(2*k+1)) = 0.276751423987223411719438512082359840225908317... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A261520 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3^k).

Original entry on oeis.org

1, 6, 36, 200, 1038, 5160, 24776, 115632, 527172, 2355998, 10349448, 44783064, 191211512, 806737800, 3367294320, 13918479872, 57020736942, 231697484304, 934399998412, 3742041461976, 14888854356840, 58881590423856, 231542984619720, 905666813058384

Offset: 0

Vaclav Kotesovec, Aug 23 2015

nonn

Convolution of A144067 and A256142.

In general, for m > 1, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(m^k), then a(n) ~ m^n * exp(2*sqrt(2*n) - 1 + c) / (sqrt(Pi) * 2^(3/4) * n^(3/4)), where c = 2 * Sum_{j>=1} 1/((2*j+1)*(m^(2*j)-1)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO]
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. 27.

Cf. A156616, A261519, A260916, A001934, A015128.

nmax = 40; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(3^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 3^n * exp(2*sqrt(2*n) - 1 + c) / (sqrt(Pi) * 2^(3/4) * n^(3/4)), where c = 2 * Sum_{j>=1} 1/((2*j+1)*(3^(2*j)-1)) = 0.0887630729103166089354170592729856346...

A300456 a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n^k).

Original entry on oeis.org

1, 2, 16, 200, 3264, 65752, 1565744, 42878432, 1324344832, 45464289482, 1715228012048, 70471268834936, 3129746696619072, 149318596196238328, 7612660420021177200, 412865831480749700928, 23725813528034949148672, 1439701175150489313314864, 91967625580609006328344400, 6167733266497532499924699672

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

nonn

			The table of coefficients of x^k in expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(n^k) begins:
n = 0: (1),  0,    0,    0,     0,       0,  ...
n = 1:  1,  (2),   4,    8,    14,      24,  ...
n = 2:  1,   4,  (16),  60,   208,     692,  ...
n = 3:  1,   6,   36, (200), 1038,    5160   ...
n = 4:  1,   8,   64,  472, (3264),  21608,  ...
n = 5:  1,  10,  100,  920,  7950,  (65752), ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..377

Cf. A015128, A252654, A261519, A261520, A270919, A270923, A270924, A292805, A300457, A300458.

Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^k))^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

a(n) ~ exp(2*sqrt(2*n) - 1) * n^(n - 3/4) / (2^(3/4)*sqrt(Pi)). - Vaclav Kotesovec, Aug 26 2019

A302237 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*(k+1)/2).

Original entry on oeis.org

1, 2, 8, 26, 76, 216, 590, 1554, 3988, 9988, 24464, 58794, 138866, 322808, 739658, 1672372, 3734848, 8245956, 18012114, 38952586, 83448832, 177194716, 373111970, 779430870, 1615995262, 3326484686, 6800794428, 13813260736, 27881653590, 55942340000, 111601021856

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

nonn

Convolution of the sequences A000294 and A028377.

N. J. A. Sloane, Transforms

Cf. A000217, A000294, A015128, A028377, A156616, A206622, A206623, A206624, A260916, A261386, A261452, A261519, A261520, A301554, A301555.

nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A000217(k).
a(n) ~ exp(2*Pi*n^(3/4)/3 + 7*Zeta(3)*sqrt(n) / (2*Pi^2) - 49*Zeta(3)^2 * n^(1/4) / (4*Pi^5) + 22411 * Zeta(3)^3 / (392*Pi^8) - Zeta(3)/(8*Pi^2) + 1/24) * Pi^(1/24) / (sqrt(A) * 2^(25/12) * n^(61/96)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2018
G.f.: A(x) = exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 - x^(2*n+1))^3) ). Cf. A000122 and A156616. - Peter Bala, Dec 23 2021

A304962 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(2^(k-1)).

Original entry on oeis.org

1, 2, 6, 18, 50, 138, 374, 994, 2610, 6778, 17414, 44346, 112034, 280970, 700038, 1733706, 4269970, 10463154, 25518198, 61962458, 149839602, 360958306, 866405702, 2072579058, 4942074082, 11748730482, 27849974598, 65837539522, 155236876018, 365125130490, 856767548022

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Convolution of the sequences A034691 and A098407.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Master's thesis, 1992. [see page 24]
N. J. A. Sloane, Transforms
Index entries for sequences related to partitions
Index entries for sequences related to compositions

Cf. A011782, A015128, A034691, A098407, A156616, A261519, A302239.

g:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      2^(d-1), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, `if`(n>1, 0, 1),
      add(b(n-i*j, i-1)*binomial(2^(i-1), j), j=0..n/i))
    end:
a:= n-> add(g(n-j)*b(j$2), j=0..n):
seq(a(n), n=0..35);  # Alois P. Heinz, May 22 2018
# Maple program to compute c(n) from a(n) or a(n) from c(n).
with(numtheory):
andrews:=proc(liste) local n,z,serie,ls,i,d,aaa;
   n:=nops(liste);
aaa:=liste;
serie:=listtoseries(aaa,z,ogf):
ls:=series(ln(serie),z,n);
   [seq(coeff(ls,z,d),d=1..n)];
   [seq(elemmobius(%,i),i=1..n-1)]
end:
swerdna:=proc(liste) local n,i,z;
  n:=nops(liste);
  series(convert([seq((1-z^i)^(-liste[i]),i=1..n)],`*`),z,n);
  [seq(coeff(%,z,i),i=0..n-1)]
end:
elemmobius:=proc(liste,d) local k,rep;
   rep:=0;
   for k in divisors(d) do
      rep:=rep+liste[k]*mobius(iquo(d,k))/iquo(d,k)
   od;
   rep
end:
# Here andrews() finds the c(n) and swerdna() finds the a(n) if the c(n) are known.
# For ordinary partitions the c(n) are [1,1,1,1,1, ...].
# Simon Plouffe, Jun 20 2018

nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(2^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A011782(k).
Euler transform of c(n) with g.f.: -x*(-2*x^2-x+2)/(-4*x^3+2*x^2+2*x-1). - Simon Plouffe, Jun 20 2018
a(n) ~ A247003 * 2^(n-1) * exp(2*sqrt(n) - 1/2 + c) / (sqrt(Pi)*n^(3/4)), where c = Sum_{k>=2} -(-1)^k / (k*(2^k-2)) = -0.207530918644117743551169251314627032... - Vaclav Kotesovec, Sep 15 2021

A302238 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^prime(k).

Original entry on oeis.org

1, 4, 14, 46, 136, 382, 1022, 2626, 6530, 15784, 37218, 85842, 194146, 431358, 943038, 2031454, 4316884, 9058662, 18787730, 38542526, 78264298, 157403290, 313712482, 619919350, 1215125262, 2363570168, 4563951858, 8751621598, 16670498062, 31553539214, 59361428202

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

nonn

Convolution of the sequences A030009 and A061152.

N. J. A. Sloane, Transforms

Cf. A000040, A015128, A030009, A061152, A156616, A206622, A206623, A206624, A260916, A261386, A261452, A261519, A261520, A301554, A301555.

nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A000040(k).

A302239 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^p(k), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, 2, 6, 16, 40, 96, 226, 512, 1140, 2488, 5336, 11270, 23494, 48356, 98438, 198338, 395846, 783136, 1536800, 2992818, 5786952, 11114950, 21213906, 40247696, 75928804, 142475644, 265985628, 494155176, 913802164, 1682338192, 3084101744, 5630853218, 10240484332, 18553818210

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

nonn

Convolution of the sequences A001970 and A261049.

N. J. A. Sloane, Transforms

Cf. A000041, A001970, A015128, A156616, A206622, A206623, A206624, A260916, A261049, A261386, A261452, A261519, A261520, A301554, A301555, A302237, A302238.

nmax = 33; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A000041(k).

cp's OEIS Frontend

A156616 G.f.: Product_{n>0} ((1+x^n)/(1-x^n))^n.

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A260916 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(Fibonacci(k)).

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A261520 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3^k).

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A300456 a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n^k).

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A302237 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*(k+1)/2).

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A304962 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(2^(k-1)).

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A302238 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^prime(k).

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A302239 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^p(k), where p(k) = number of partitions of k (A000041).

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