A156616 -id:A156616 - cp's OEIS frontend

A261649 Expansion of Product_{k>=0} ((1+x^(3k+1))/(1-x^(3k+1)))^2.

1, 4, 8, 12, 20, 36, 56, 80, 120, 180, 252, 348, 492, 680, 912, 1228, 1652, 2180, 2856, 3744, 4860, 6256, 8044, 10284, 13048, 16520, 20848, 26140, 32672, 40756, 50596, 62576, 77256, 95060, 116540, 142592, 174036, 211736, 257056, 311448, 376332, 453764, 546160

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

Cf. A015128, A156616, A080054, A007096, A261610, A261651.

nmax=60; CoefficientList[Series[Product[((1+x^(3*k+1))/(1-x^(3*k+1)))^2,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(2*n/3)) * Gamma(1/3)^2 / (2^(7/4) * 3^(5/12) * Pi^(4/3) * n^(7/12)).

A261651 Expansion of Product_{k>=0} ((1+x^(3k+1))/(1-x^(3k+1)))^3.

Original entry on oeis.org

1, 6, 18, 38, 72, 138, 254, 432, 708, 1154, 1836, 2826, 4288, 6456, 9552, 13902, 20070, 28722, 40614, 56916, 79242, 109448, 149988, 204318, 276672, 372288, 498264, 663602, 879252, 1159470, 1522564, 1990788, 2592162, 3362638, 4346244, 5597100, 7183792, 9191004

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

Cf. A015128, A156616, A080054, A007096, A261610, A261649.

nmax=60; CoefficientList[Series[Product[((1+x^(3*k+1))/(1-x^(3*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(n)) * Gamma(1/3)^3 / (4 * Pi^2 * sqrt(3*n)).

A261652 Expansion of Product_{k>=0} ((1+x^(4k+1))/(1-x^(4k+1)))^3.

Original entry on oeis.org

1, 6, 18, 38, 66, 108, 182, 306, 486, 728, 1068, 1578, 2318, 3312, 4614, 6388, 8862, 12192, 16488, 22038, 29400, 39156, 51702, 67554, 87810, 113982, 147384, 189200, 241446, 307356, 390408, 493662, 621006, 778712, 974628, 1216284, 1511756, 1872840, 2315538

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

In general, if j > 0, a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} ((1 + x^(a*k+b))/(1 - x^(a*k+b)))^j, then a(n) ~ Gamma(b/a)^j * 2^(j/2 - 3/2 - 2*b*j/a) * a^(-j/4 - 1/4 + b*j/(2*a)) * exp(Pi*sqrt(j*n/a)) * j^(1/4 - j/4 + b*j/(2*a)) * Pi^(b*j/a - j) * n^(j/4 - 3/4 - b*j/(2*a)).

Cf. A015128 (a=1, b=1, j=1), A156616.
Cf. A080054 (a=2, b=1, j=1), A007096 (a=2, b=1, j=2), A261647 (a=2, b=1, j=3), A014969 (a=2, b=1, j=4), A261648 (a=2, b=1, j=5), A014970 (a=2, b=1, j=6), A014972 (a=2, b=1, j=8), A103261 (a=2, b=1, j=10).
Cf. A261610 (a=3, b=1, j=1), A261649 (a=3, b=1, j=2), A261651 (a=3, b=1, j=3).
Cf. A261611 (a=4, b=1, j=1), A261650 (a=4, b=1, j=2), A261652 (a=4, b=1, j=3).

nmax=60; CoefficientList[Series[Product[((1+x^(4*k+1))/(1-x^(4*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(3*n)/2) * 2^(1/4) * Gamma(1/4)^3 / (8 * 3^(1/8) * Pi^(9/4) * n^(3/8)).

A291666 Expansion of Product_{k>=1} ((1 + x^(k^2)) / (1 - x^(k^2)))^(k^2).

Original entry on oeis.org

1, 2, 2, 2, 10, 18, 18, 18, 50, 100, 118, 118, 206, 438, 582, 582, 806, 1606, 2344, 2506, 3122, 5322, 8202, 9498, 11130, 16844, 26110, 32272, 37018, 52274, 78018, 100098, 115986, 155026, 223190, 291674, 345132, 439518, 618734, 811790, 972846, 1204190, 1653726

Offset: 0

Vaclav Kotesovec, Aug 29 2017

nonn

Convolution of A291649 and A291655.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001156, A033461, A103265, A156616, A206622, A291649, A291655, A291667, A291721.

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

log(a(n)) ~ 5 * Pi^(1/5) * ((8-sqrt(2)) * Zeta(5/2))^(2/5) * n^(3/5) / (4*3^(3/5)).

A291721 Expansion of Product_{k>=1} ((1 + x^(k^3))/(1 - x^(k^3)))^(k^3).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 18, 34, 34, 34, 34, 34, 34, 34, 162, 290, 290, 290, 290, 290, 290, 290, 978, 1666, 1666, 1720, 1774, 1774, 1774, 1774, 4590, 7406, 7406, 8270, 9134, 9134, 9134, 9134, 18558, 27982, 27982, 34894, 41806, 41806, 41806, 41806, 68814, 95822

Offset: 0

Vaclav Kotesovec, Aug 30 2017

nonn

Convolution of A291692 and A291720.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A156616, A291666, A291692, A291720.

nmax = 100; CoefficientList[Series[Product[((1 + x^(k^3))/(1 - x^(k^3)))^(k^3), {k, 1, nmax}], {x, 0, nmax}], x]

log(a(n)) ~ 7 * ((2^(7/3)-1) * Gamma(1/3) * Zeta(7/3))^(3/7) * n^(4/7) / (2^(12/7) * 3^(9/7)).

A216406 G.f.: Product_{n>=1} ((1-x^n)/(1+x^n))^(2*n).

Original entry on oeis.org

1, -4, 0, 8, 16, -8, -48, -56, 0, 116, 256, 264, -32, -648, -1296, -1392, -352, 2040, 5200, 7368, 6112, -784, -13744, -29304, -39648, -33804, -1376, 60368, 139552, 205304, 210208, 103432, -146528, -521744, -928480, -1190000, -1069904, -339720, 1110864, 3146640, 5278624

Offset: 0

Paul D. Hanna, Sep 06 2012

sign

The number of contiguous signs seems to increase in proportion to the square-root of the number of terms.
Compare the g.f. to the Jacobi theta_4 series identity:
exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*x^n/n ) = 1 + 2*Sum_{n>=1} (-x)^(n^2).

			G.f.: A(x) = 1 - 4*x + 8*x^3 + 16*x^4 - 8*x^5 - 48*x^6 - 56*x^7 + 116*x^9 +...
where the g.f. equals the infinite product:
A(x) = (1-x)^2/(1+x)^2 * (1-x^2)^4/(1+x^2)^4 * (1-x^3)^6/(1+x^3)^6 * (1-x^4)^8/(1+x^4)^8 * (1-x^5)^10/(1+x^5)^10 *...
The logarithm of the g.f. is illustrated by:
-log(A(x)) = 4*x + 16*x^2/2 + 40*x^3/3 + 64*x^4/4 + 104*x^5/5 + 160*x^6/6 + 200*x^7/7 + 256*x^8/8 +...+ 4*A076577(n)*x^n/n +...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from Paul D. Hanna)

Cf. A156616, A076577, A001157 (sigma_2), A261386.

{a(n)=polcoeff(exp(sum(m=1, n+1, -(sigma(2*m,2)-sigma(m,2))*x^m/m+x*O(x^n))), n)}

{a(n)=polcoeff(prod(m=1,n,((1-x^m)/(1+x^m +x*O(x^n)))^(2*m)), n)}
for(n=0, 100, print1(a(n), ", "))

G.f.: exp( Sum_{n>=1} -(sigma_2(2*n) - sigma_2(n))*x^n/n ) where sigma_2(n) = sum of squares of divisors of n.

A261650 Expansion of Product_{k>=0} ((1+x^(4k+1))/(1-x^(4k+1)))^2.

Original entry on oeis.org

1, 4, 8, 12, 16, 24, 40, 60, 80, 104, 144, 204, 272, 344, 440, 584, 768, 968, 1200, 1516, 1936, 2424, 2968, 3644, 4528, 5596, 6800, 8216, 10000, 12184, 14688, 17564, 21056, 25320, 30272, 35912, 42576, 50616, 60024, 70728, 83136, 97896, 115200, 134924, 157504

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

Cf. A015128, A156616, A080054, A007096, A261611, A261652.

nmax=60; CoefficientList[Series[Product[((1+x^(4*k+1))/(1-x^(4*k+1)))^2,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(n/2)) * 2^(3/2) * Gamma(1/4)^2 / (16 * Pi^(3/2) * sqrt(n)).

A294780 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*(k-1)/2).

Original entry on oeis.org

1, 0, 2, 6, 14, 32, 74, 166, 370, 810, 1736, 3682, 7718, 15976, 32754, 66508, 133794, 266948, 528424, 1038178, 2025456, 3925360, 7559298, 14470162, 27540598, 52130440, 98159832, 183905636, 342896254, 636384748, 1175823512, 2163221030, 3963353706, 7232529308

Offset: 0

Vaclav Kotesovec, Nov 08 2017

nonn

Convolution of A027999 and A258349.

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

Cf. A027999, A156616, A206622, A258349, A294779.

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

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) * sqrt(A) / (2^(23/12) * Pi^(1/24) * n^(59/96)), where A is the Glaisher-Kinkelin constant A074962.

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

cp's OEIS Frontend

A261649 Expansion of Product_{k>=0} ((1+x^(3*k+1))/(1-x^(3*k+1)))^2.

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A261651 Expansion of Product_{k>=0} ((1+x^(3*k+1))/(1-x^(3*k+1)))^3.

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A261652 Expansion of Product_{k>=0} ((1+x^(4*k+1))/(1-x^(4*k+1)))^3.

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A291666 Expansion of Product_{k>=1} ((1 + x^(k^2)) / (1 - x^(k^2)))^(k^2).

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A291721 Expansion of Product_{k>=1} ((1 + x^(k^3))/(1 - x^(k^3)))^(k^3).

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A216406 G.f.: Product_{n>=1} ((1-x^n)/(1+x^n))^(2*n).

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A261650 Expansion of Product_{k>=0} ((1+x^(4*k+1))/(1-x^(4*k+1)))^2.

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A294780 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*(k-1)/2).

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

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A261649 Expansion of Product_{k>=0} ((1+x^(3k+1))/(1-x^(3k+1)))^2.

A261651 Expansion of Product_{k>=0} ((1+x^(3k+1))/(1-x^(3k+1)))^3.

A261652 Expansion of Product_{k>=0} ((1+x^(4k+1))/(1-x^(4k+1)))^3.

A261650 Expansion of Product_{k>=0} ((1+x^(4k+1))/(1-x^(4k+1)))^2.