A338673 Expansion of Product_{k>=1} 1 / (1 - 4^(k-1)*x^k).

1, 1, 5, 21, 101, 421, 2021, 8421, 39397, 167397, 766437, 3244517, 14881253, 62804453, 283415013, 1210159589, 5401907685, 22966866405, 102497423845, 435085808101, 1925197238757, 8215432696293, 36068400468453, 153579729097189, 674546796630501, 2866238341681637, 12508012102193637

Offset

0,3

Links

Table of n, a(n) for n=0..26.

Formula

a(n) = Sum_{k=0..n} p(n,k) * 4^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(3) * polylog(2, 1/4)^(1/4) * 4^(n - 1/2) * exp(2*sqrt(polylog(2, 1/4)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

Mathematica

nmax = 26; CoefficientList[Series[Product[1/(1 - 4^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 4^(n - k), {k, 0, n}], {n, 0, 26}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 4^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 26}]

Crossrefs

Cf. A008284, A075900, A246936, A300579, A338674, A338675, A338676, A338677, A338678, A338679.

Sequence in context: A017968 A017969 A050897 * A199215 A204061 A362556

Adjacent sequences: A338670 A338671 A338672 * A338674 A338675 A338676

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 23 2021

Status

approved