A258798 -id:A258798 - cp's OEIS frontend

A258797 a(n) = [x^n] Product_{k=1..n} (1+x^k)^2 / x^k.

1, 1, 2, 6, 16, 51, 166, 554, 1896, 6595, 23212, 82582, 296393, 1071738, 3900696, 14278074, 52526972, 194108087, 720197524, 2681854490, 10019539112, 37545876368, 141080872362, 531457445806, 2006678785762, 7593123695669, 28789152013570, 109356019134584

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

a(n) is half the number of subsets of {-n..n} whose sum is n. - Ilya Gutkovskiy, Jul 09 2025

Alois P. Heinz, Table of n, a(n) for n = 0..555

Cf. A022567, A047653, A258798, A258799.

b:= proc(n, s) option remember; `if`(n*(n+1)/2 b(n$2):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 14 2025

Table[SeriesCoefficient[Product[(1+x^k)^2/x^k, {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[Product[1+x^k, {k, 1, n}]^2, {x, 0, n*(n+3)/2}], {n, 0, 30}]

a(n) ~ sqrt(3) * 4^n / (sqrt(Pi) * n^(3/2)).

A258799 a(n) = [x^n] Product_{k=1..n} (1+x^k)^3 / x^(2*k).

Original entry on oeis.org

1, 1, 3, 13, 61, 324, 1800, 10340, 60969, 366486, 2237120, 13829487, 86394782, 544547651, 3458637273, 22113504345, 142212705879, 919294844898, 5969839457411, 38927450022860, 254776529381625, 1673102335692514, 11020847332241873, 72798664086854460

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

Cf. A258797, A258798.

Table[SeriesCoefficient[Product[(1+x^k)^3/x^(2*k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[Product[1+x^k, {k, 1, n}]^3, {x, 0, n*(n+2)}], {n, 0, 30}]
(* A program to compute the constant d *) (1+r)^3/r /.FindRoot[Log[1+r]/Log[r] + (PolyLog[2,-r] + Pi^2/12) / Log[r]^2 == 1/6, {r, E}, WorkingPrecision->100]

a(n) ~ c * d^n / n^(3/2), where d = 7.036711302278424796297167109247361745558645910729132828752853658917..., c = 0.282321145891... .

cp's OEIS Frontend

A258797 a(n) = [x^n] Product_{k=1..n} (1+x^k)^2 / x^k.

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