A343360 Expansion of Product_{k>=1} (1 + x^k)^(3^(k-1)).

1, 1, 3, 12, 39, 138, 469, 1603, 5427, 18372, 61869, 207909, 696537, 2328039, 7762266, 25826142, 85749969, 284171598, 940027872, 3104280885, 10234808334, 33692547249, 110753171784, 363561071175, 1191860487561, 3902350627434, 12761565487173, 41685086306917, 136012008938158

Offset

0,3

Links

Table of n, a(n) for n=0..28.

Formula

a(n) ~ exp(2*sqrt(n/3) - 1/6 - c/3) * 3^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (3^(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(3^(i-1), j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..28);  # Alois P. Heinz, Apr 12 2021

Mathematica

nmax = 28; CoefficientList[Series[Product[(1 + x^k)^(3^(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 3^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 28}]

Prog

(PARI) seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(3^(k-1))))} \\ Andrew Howroyd, Apr 12 2021

Crossrefs

Cf. A098407, A104460, A256142, A343361, A343362, A343363, A343364, A343365, A343366.

Sequence in context: A330169 A290906 A110153 * A183366 A122994 A271218

Adjacent sequences: A343357 A343358 A343359 * A343361 A343362 A343363

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 12 2021

Status

approved