A266942 -id:A266942 - cp's OEIS frontend

A266941 Expansion of Product_{k>=1} 1 / (1 - k*x^k)^k.

1, 1, 5, 14, 42, 103, 289, 690, 1771, 4206, 10142, 23449, 54786, 123528, 279480, 619206, 1366405, 2969071, 6425534, 13727775, 29187555, 61439660, 128620370, 267044222, 551527679, 1130806020, 2306746335, 4676096006, 9432394144, 18920266428, 37776372312

Offset: 0

Vaclav Kotesovec, Jan 06 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..6086 (terms 0..1000 from Vaclav Kotesovec)

Cf. A006906, A265758, A266891, A266942, A266964, A266971.

nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; s = 1 - x; Do[s *= Sum[Binomial[k, j]*(-1)^j*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}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2018 *)

From Vaclav Kotesovec, Jan 08 2016: (Start)
a(n) ~ c * n^2 * 3^(n/3), where
c = 3278974684037157122864203.021982619109776972432419491714093... if mod(n,3)=0
c = 3278974684037157122864202.999526122508793149896683112820555... if mod(n,3)=1
c = 3278974684037157122864203.001231135511323719311281438384212... if mod(n,3)=2
(End)
In closed form, a(n) ~ (Product_{k>=4}((1 - k/3^(k/3))^(-k)) / ((1 - 2/3^(2/3))^2 * (1 - 1/3^(1/3))) + Product_{k>=4}((1 - (-1)^(2*k/3)*k/3^(k/3))^(-k)) / ((-1)^(2*n/3) * ((1 + 2*(-1)^(1/3)/3^(2/3))^2 * (1 - (-1)^(2/3)/3^(1/3)))) + Product_{k>=4}((1 - (-1)^(4*k/3)*k/3^(k/3))^(-k)) / ((-1)^(4*n/3) * ((1 + (-1/3)^(1/3)) * (1 - 2*(-1/3)^(2/3))^2))) * 3^(n/3) * n^2 / 54. - Vaclav Kotesovec, Apr 24 2017
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (Sum_{d|k} d^(2+k/d)) * a(n-k) for n > 0. - Seiichi Manyama, Nov 02 2017

A305745 Expansion of Product_{k>=1} ((1 - kx^k) / (1 + kx^k))^k.

Original entry on oeis.org

1, -2, -6, -4, 22, 72, 84, -32, -474, -1310, -1728, 60, 6420, 18712, 31080, 24992, -34074, -186468, -430138, -650612, -496296, 687120, 3599652, 8413968, 13374148, 12772246, -3910080, -50592280, -136089520, -244815336, -309079848, -176916784, 391358838

Offset: 0

Seiichi Manyama, Jun 09 2018

sign

Robert Israel, Table of n, a(n) for n = 0..6234

Cf. A266942, A266964, A266971, A292317.

N:= 50: # for a(0) .. a(N)
f:= mul(((1-k*x^k)/(1+k*x^k))^k,k=1..N):
S:= series(f,x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Mar 01 2024

N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-k*x^k)/(1+k*x^k))^k))

Convolution of A266964 and A266971.

Convolution inverse of A266942.

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

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A266941 Expansion of Product_{k>=1} 1 / (1 - k*x^k)^k.

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A305745 Expansion of Product_{k>=1} ((1 - kx^k) / (1 + kx^k))^k.

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

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cp's OEIS Frontend

A266941 Expansion of Product_{k>=1} 1 / (1 - k*x^k)^k.

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A305745 Expansion of Product_{k>=1} ((1 - k*x^k) / (1 + k*x^k))^k.

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Maple

PARI

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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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A305745 Expansion of Product_{k>=1} ((1 - kx^k) / (1 + kx^k))^k.

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