A258799 -id:A258799

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A258797

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..27.

FORMULA

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

MATHEMATICA

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}]

CROSSREFS

Cf. A258798, A258799.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Jun 10 2015

STATUS

approved

A258798

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

+10
2

1, 3, 12, 62, 327, 1851, 10802, 64440, 391218, 2408001, 14989608, 94197594, 596756374, 3807010920, 24435261432, 157681777148, 1022391454116, 6657413851086, 43517229086467, 285447137446989, 1878287880309900, 12395149957521672, 82014499806039711

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..22.

FORMULA

a(n) ~ c * d^n / n^(3/2), where d = 7.036711302278424796297167109247361745558645910729132828752853658917..., c = 2.3254811458... .

MATHEMATICA

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

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

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