A285674 -id:A285674 - cp's OEIS frontend

A294586 Expansion of Product_{k>=1} 1/(1 - k^2*x^k)^(k^2).

1, 1, 17, 98, 514, 2435, 12752, 58849, 277362, 1243056, 5523734, 23889860, 102176581, 427458488, 1768064752, 7197695011, 28955246228, 114977761216, 451686925462, 1754581791860, 6749143188662, 25707194720502, 97041994691555, 363121143230292

Offset: 0

Seiichi Manyama, Nov 03 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..2889

Column k=2 of A294585.

Cf. A077335, A285241, A285674, A294584.

nmax = 30; CoefficientList[Series[Product[1/(1 - k^2*x^k)^(k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 11 2017 *)
nmax = 30; s = 1 - x; Do[s *= Sum[Binomial[k^2, j]*(-1)^j*k^(2*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, Nov 12 2017 *)

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-k^2*x^k)^k^2))

From Vaclav Kotesovec, Nov 14 2017: (Start)
a(n) ~ c * 3^(2*n/3) * n^8, where
if mod(n,3)=0 then c = 350793443467906700358779160929996923840677857044\
13786172.61998576944425459411592809123023259309183199454386580509531344\
26216683391121761062030679551011342614958936988089343473390138...
if mod(n,3)=1 then c = 350793443467906700358779160929996923840677857044\
13786172.61998576943431618172412821798685989333734080090574886961583670\
65437558779530384541992249698997443314123905740649930258416583...
if mod(n,3)=2 then c = 350793443467906700358779160929996923840677857044\
13786172.61998576943586440772541471067224229278174424709431922476448338\
37991534958575385658058309282842532811502400165735702386411333...
In closed form, a(n) ~ ((Product_{k>=4} ((1 - k^2 / 3^(2*k/3))^(-k^2))) / ((1 - 1/3^(2/3)) * (1 - 4/3^(4/3))^4) + (Product_{k>=4} ((1 - (-1)^(2*k/3) * k^2 / 3^(2*k/3))^(-k^2))) / ((-1)^(2*n/3) * (1 + 4/3 * (-1/3)^(1/3))^4 * (1 - (-1/3)^(2/3))) + (Product_{k>=4} ((1 - (-(-1)^(1/3))^k * k^2 / 3^(2*k/3))^(-k^2))) / ((-(-1)^(1/3))^n * (1 + (-1)^(1/3) / 3^(2/3)) * (1 - 4*(-1)^(2/3) / 3^(4/3))^4)) * 3^(2*n/3) * n^8 / 793618560. - Vaclav Kotesovec, Nov 14 2017 (End)

A285737 Expansion of Product_{k>=1} (1 + k^2*x^k)^k.

Original entry on oeis.org

1, 1, 8, 35, 107, 421, 1312, 4474, 13622, 43977, 130473, 388025, 1146640, 3265446, 9352424, 26033637, 72144351, 196664848, 532768901, 1422725368, 3768251677, 9893857617, 25709347054, 66367179293, 169754459790, 431237516979, 1086813719408, 2722241654623

Offset: 0

Vaclav Kotesovec, Apr 25 2017

nonn

Cf. A026007, A092484, A266891, A285674.

nmax=40; CoefficientList[Series[Product[(1+k^2*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

A294582 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j^k*x^j)^j.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 9, 14, 13, 1, 1, 17, 36, 42, 24, 1, 1, 33, 98, 148, 103, 48, 1, 1, 65, 276, 546, 489, 289, 86, 1, 1, 129, 794, 2068, 2467, 1959, 690, 160, 1, 1, 257, 2316, 7962, 12969, 14281, 6326, 1771, 282, 1, 1, 513, 6818, 30988, 70243

Offset: 0

Seiichi Manyama, Nov 02 2017

			Square array begins:
    1,  1,   1,   1,    1, ...
    1,  1,   1,   1,    1, ...
    3,  5,   9,  17,   33, ...
    6, 14,  36,  98,  276, ...
   13, 42, 148, 546, 2068, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A000219, A266941, A285674.
Rows n=0-1 give A000012.
Cf. A292193, A294580.

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(2+k*j/d)) * A(n-j,k) for n > 0.

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A294586 Expansion of Product_{k>=1} 1/(1 - k^2*x^k)^(k^2).

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A285737 Expansion of Product_{k>=1} (1 + k^2*x^k)^k.

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Mathematica

A294582 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j^k*x^j)^j.

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