A156616 -id:A156616 - cp's OEIS frontend

A318579 Expansion of Product_{i>=1, j>=1} ((1 + x^(ij))/(1 - x^(ij)))^(i*j).

1, 2, 10, 30, 98, 270, 786, 2046, 5418, 13556, 33726, 81002, 192902, 447562, 1027990, 2316750, 5165398, 11345298, 24668952, 52972902, 112688802, 237193354, 494933514, 1023238806, 2098662698, 4269141516, 8620916966, 17280687472, 34405835066, 68044209950, 133732805458

Offset: 0

Ilya Gutkovskiy, Aug 29 2018

nonn

Convolution of A280540 and A280541.

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

Cf. A000005, A038040, A156616, A280540, A280541, A301554, A301555.

a:=series(mul(mul(((1+x^(i*j))/(1-x^(i*j)))^(i*j),j=1..100),i=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 02 2019

nmax = 30; CoefficientList[Series[Product[Product[((1 + x^(i j))/(1 - x^(i j)))^(i j), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k DivisorSigma[0, k]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(1 - (-1)^(k/d)) d^2 DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 30}]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*tau(k)), where tau(k) = number of divisors of k (A000005).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (1 - (-1)^(k/d))*d^2*tau(d) ) * x^k/k).
log(a(n)) ~ 3^(2/3) * (7*Zeta(3))^(1/3) * log(n)^(1/3) * n^(2/3) / 2^(4/3). - Vaclav Kotesovec, Sep 03 2018

A361008 G.f.: Product_{k >= 0} ((1 + x^(2k+1)) / (1 - x^(2k+1)))^k.

Original entry on oeis.org

1, 0, 0, 2, 0, 4, 2, 6, 8, 10, 20, 18, 42, 40, 78, 92, 140, 192, 258, 382, 480, 728, 902, 1334, 1698, 2404, 3148, 4292, 5742, 7608, 10304, 13430, 18192, 23592, 31720, 41144, 54766, 71188, 93762, 122156, 159420, 207820, 269380, 350726, 452434, 587520, 755446

Offset: 0

Vaclav Kotesovec, Apr 20 2023

nonn

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

Cf. A080054, A291697, A263149, A263150.

Cf. A156616, A270919.

Table[SeriesCoefficient[Product[((1 + x^(2*k + 1))/(1 - x^(2*k + 1)))^k, {k, 0, n}], {x, 0, n}], {n, 0, 50}]

a(n) ~ sqrt(A/(3*Pi)) * (7*zeta(3))^(11/72) * exp(3*(7*zeta(3))^(1/3) * n^(2/3)/4 - Pi^2 * n^(1/3)/(8*(7*zeta(3))^(1/3)) - 1/24 - Pi^4/(1344*zeta(3))) / (2^(3/4) * n^(47/72)), where A = A074962 is the Glaisher-Kinkelin constant.

A265015 a(n) = A015128(n)^n.

Original entry on oeis.org

1, 2, 16, 512, 38416, 7962624, 4096000000, 4398046511104, 10000000000000000, 48717667557975775744, 451730952053751361306624, 7982572438812891719395180544, 268637376395543538746286686601216, 16132732437821617561429013924830773248

Offset: 0

Vaclav Kotesovec, Dec 05 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..95

Cf. A015128, A133018, A156616, A265094.

nmax = 15; A015128 = Rest[CoefficientList[Series[Product[(1+x^k)/(1-x^k), {k,1,nmax}], {x,0,nmax}], x]]; Flatten[{1, Table[A015128[[n]]^n, {n,1,nmax}]}]

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

A301625 G.f. A(x) satisfies: A(x) = Product_{k>=1} ((1 + x^kA(x)^k)/(1 - x^kA(x)^k))^k.

Original entry on oeis.org

1, 2, 10, 60, 398, 2820, 20892, 159868, 1253758, 10024070, 81400672, 669532924, 5566386324, 46701736772, 394910202608, 3362210548344, 28797181196766, 247955463799812, 2145088563952510, 18636002388075260, 162523319555310664, 1422259430668179592, 12485554521209720492, 109922263517662775292

Offset: 0

Ilya Gutkovskiy, Mar 24 2018

nonn

			G.f. A(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 398*x^4 + 2820*x^5 + 20892*x^6 + 159868*x^7 + 1253758*x^8 + ...
G.f. A(x) satisfies: A(x) = ((1 + x*A(x)) * (1 + x^2*A(x)^2)^2 * (1 + x^3*A(x)^3)^3 * ...)/((1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * ...).
log(A(x)) = 2*x + 16*x^2/2 + 128*x^3/3 + 1056*x^4/4 + 8952*x^5/5 + 77200*x^6/6 + 673948*x^7/7 + 5937792*x^8/8 + ... + A270924(n)*x^n/n + ...

Cf. A006195, A066398, A109085, A156616, A181315, A270924, A278428, A301455, A301456, A301624.

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).

A318764 Expansion of Product_{i>=1, j>=1, k>=1} ((1 + x^(ijk))/(1 - x^(ijk)))^(ijk).

Original entry on oeis.org

1, 2, 14, 44, 182, 548, 1932, 5632, 17654, 49872, 145020, 395256, 1090044, 2876424, 7606024, 19503312, 49850790, 124543772, 309436980, 755268832, 1831194724, 4376807896, 10387118328, 24359228520, 56720659372, 130737105940, 299256890672, 678941040784

Offset: 0

Vaclav Kotesovec, Sep 03 2018

nonn

Convolution of A318413 and A318414.

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

Cf. A318413, A318414.
Cf. A156616, A318579.
Cf. A015128, A305050.

nmax = 40; CoefficientList[Series[Product[Product[Product[((1 + x^(i*j*k))/(1 - x^(i*j*k)))^(i*j*k), {i, 1, nmax/j/k}], {j, 1, nmax/k}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k*Sum[DivisorSigma[0, d], {d, Divisors[k]}]), {k, 1, nmax}], {x, 0, nmax}], x]

Conjecture: log(a(n)) ~ (21*Zeta(3))^(1/3) * log(n)^(2/3) * n^(2/3) / 2^(5/3).

A296048 Expansion of e.g.f. Product_{k>=1} ((1 - x^k)/(1 + x^k))^(1/k).

Original entry on oeis.org

1, -2, 2, -4, 32, -128, 496, -2336, 29312, -395776, 3194624, -21951488, 277270528, -4027191296, 38850203648, -739834458112, 19460560584704, -299971773661184, 3169121209090048, -51853341314514944, 1234704403684130816, -30653318499154788352, 658369600764729884672, -10809496145754051313664

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

sign

Cf. A001227, A028342, A028343, A054844, A156616, A168243, A285675, A294356, A295792.

a:=series(mul(((1-x^k)/(1+x^k))^(1/k),k=1..100),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

nmax = 23; CoefficientList[Series[Product[((1 - x^k)/(1 + x^k))^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[-2 Sum[Total[Mod[Divisors[k], 2] x^k]/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(-2*Sum_{k>=1} A001227(k)*x^k/k).

E.g.f.: exp(-Sum_{k>=1} A054844(k)*x^k/k).

A300412 a(n) = [x^n] Product_{k>=1} ((1 + nx^k)/(1 - nx^k))^k.

Original entry on oeis.org

1, 2, 16, 144, 1376, 15800, 210816, 3333372, 61688448, 1318588146, 32004369200, 869282342632, 26099925704928, 857736429098848, 30605729417479104, 1177841009504482200, 48614265201514729984, 2141639401723095243324, 100282931820560447963568, 4973060138191518242569120

Offset: 0

Ilya Gutkovskiy, Mar 05 2018

nonn

			The table of coefficients of x^k in expansion of Product_{k>=1} ((1 + n*x^k)/(1 - n*x^k))^k begins:
n = 0: (1),  0,   0,    0,     0,       0,  ...
n = 1:  1,  (2),  6,   16,    38,      88,  ...
n = 2:  1,   4, (16),  60,   192,     596,  ...
n = 3:  1,   6,  30, (144),  582,    2280,  ...
n = 4:  1,   8,  48,  280, (1376),   6568,  ...
n = 5:  1,  10,  70,  480,  2790,  (15800), ...

Cf. A156616, A261563, A266942, A270919, A270924, A292419, A298985, A298987.

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

a(n) ~ 2 * n^n * (1 + 4/n + 14/n^2 + 44/n^3 + 124/n^4 + 328/n^5 + 824/n^6 + 1980/n^7 + 4590/n^8 + 10320/n^9 + 22584/n^10 + ...), for coefficients see A261451. - Vaclav Kotesovec, Mar 05 2018

A321389 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k^k).

Original entry on oeis.org

1, 2, 10, 72, 670, 7896, 113532, 1938948, 38463150, 869969602, 22098936536, 622728174288, 19271479902324, 649553475002720, 23680210649058960, 928276725059295192, 38931910620358040382, 1739307894106738293052, 82457731356894087128054, 4134332188240252347401752, 218571692793801915329820184

Offset: 0

Ilya Gutkovskiy, Nov 08 2018

nonn

Convolution of A023880 and A261053.

N. J. A. Sloane, Transforms

Cf. A023880, A156616, A206622, A206623, A206624, A261053.

a:=series(mul(((1+x^k)/(1-x^k))^(k^k),k=1..100),x=0,21): seq(coeff(a,x,n),n=0..20); # Paolo P. Lava, Apr 02 2019

nmax = 20; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[((-1)^(k/d + 1) + 1) d^(d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]

seq(n)={Vec(exp(sum(k=1, n, sumdiv(k,d, ((-1)^(k/d+1) + 1)*d^(d+1) ) * x^k/k) + O(x*x^n)))} \\ Andrew Howroyd, Nov 09 2018

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} ((-1)^(k/d+1) + 1)*d^(d+1) ) * x^k/k).

a(n) ~ 2 * n^n * (1 + 2*exp(-1)/n + (exp(-1) + 10*exp(-2))/n^2). - Vaclav Kotesovec, Nov 09 2018

cp's OEIS Frontend

A318579 Expansion of Product_{i>=1, j>=1} ((1 + x^(i*j))/(1 - x^(i*j)))^(i*j).

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A361008 G.f.: Product_{k >= 0} ((1 + x^(2*k+1)) / (1 - x^(2*k+1)))^k.

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A265015 a(n) = A015128(n)^n.

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A301625 G.f. A(x) satisfies: A(x) = Product_{k>=1} ((1 + x^k*A(x)^k)/(1 - x^k*A(x)^k))^k.

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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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A318764 Expansion of Product_{i>=1, j>=1, k>=1} ((1 + x^(i*j*k))/(1 - x^(i*j*k)))^(i*j*k).

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A296048 Expansion of e.g.f. Product_{k>=1} ((1 - x^k)/(1 + x^k))^(1/k).

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A300412 a(n) = [x^n] Product_{k>=1} ((1 + n*x^k)/(1 - n*x^k))^k.

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A318579 Expansion of Product_{i>=1, j>=1} ((1 + x^(ij))/(1 - x^(ij)))^(i*j).

A361008 G.f.: Product_{k >= 0} ((1 + x^(2k+1)) / (1 - x^(2k+1)))^k.

A301625 G.f. A(x) satisfies: A(x) = Product_{k>=1} ((1 + x^kA(x)^k)/(1 - x^kA(x)^k))^k.

A318764 Expansion of Product_{i>=1, j>=1, k>=1} ((1 + x^(ijk))/(1 - x^(ijk)))^(ijk).

A300412 a(n) = [x^n] Product_{k>=1} ((1 + nx^k)/(1 - nx^k))^k.