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A343361
Expansion of Product_{k>=1} (1 + x^k)^(4^(k-1)).
7
1, 1, 4, 20, 86, 390, 1724, 7644, 33697, 148401, 651584, 2855840, 12491276, 54540636, 237733768, 1034610232, 4495832776, 19508749928, 84540638312, 365888222552, 1581630245756, 6829047398156, 29453496620000, 126898489491904, 546183557447366, 2348560270762006, 10089340886428928
OFFSET
0,3
FORMULA
a(n) ~ exp(sqrt(n) - 1/8 - c/4) * 2^(2*n - 3/2) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (4^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021
MAPLE
h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(h(n-i*j, i-1)*binomial(4^(i-1), j), j=0..n/i)))
end:
a:= n-> h(n$2):
seq(a(n), n=0..26); # Alois P. Heinz, Apr 12 2021
MATHEMATICA
nmax = 26; CoefficientList[Series[Product[(1 + x^k)^(4^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 4^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 26}]
PROG
(PARI) seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(4^(k-1))))} \\ Andrew Howroyd, Apr 12 2021
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 12 2021
STATUS
approved