A294584 -id:A294584 - cp's OEIS frontend

A294583 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j^k*x^j)^(j^k).

1, 1, -1, 1, -1, -1, 1, -1, -4, 0, 1, -1, -16, -5, 0, 1, -1, -64, -65, -3, 1, 1, -1, -256, -665, -79, 23, 0, 1, -1, -1024, -6305, -1575, 831, 44, 1, 1, -1, -4096, -58025, -28255, 33335, 4789, 104, 0, 1, -1, -16384, -527345, -481623, 1323807, 411664, 15099, 70, 0

Offset: 0

Seiichi Manyama, Nov 03 2017

			Square array begins:
    1,  1,   1,     1,      1, ...
   -1, -1,  -1,    -1,     -1, ...
   -1, -4, -16,   -64,   -256, ...
    0, -5, -65,  -665,  -6305, ...
    0, -3, -79, -1575, -28255, ...

Seiichi Manyama, Antidiagonals n = 0..113, flattened

Columns k=0..2 give A010815, A266964, A294584.
Rows n=0..1 give A000012, (-1)*A000012.
Cf. A294585.

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

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

Original entry on oeis.org

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)

cp's OEIS Frontend

A294583 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j^k*x^j)^(j^k).

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