A366203 a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(3*n,n-k-1) * (n-3)^k.

1, 2, 12, 156, 3507, 115692, 5066364, 276943568, 18152243967, 1387267590540, 121106707350928, 11889022355301672, 1296359140925188212, 155440199716271334648, 20327081449263918542412, 2879054747404226046119448, 439060192463001381367975215, 71727764882350305085962745740

Offset

1,2

Comments

a(n) is the coefficient of x^n in expansion of series reversion of g.f. for n-gonal numbers (with signs).

Links

Table of n, a(n) for n=1..18.

Formula

a(n) = [x^n] Series_Reversion( x * (1 - (n - 3) * x) / (1 + x)^3 ).

Mathematica

Unprotect[Power]; 0^0 = 1; Table[1/n Sum[Binomial[n + k - 1, k] Binomial[3 n, n - k - 1] (n - 3)^k, {k, 0, n - 1}], {n, 1, 18}]
Table[Binomial[3 n, n - 1] Hypergeometric2F1[1 - n, n, 2 (n + 1), 3 - n]/n, {n, 1, 18}]
Table[SeriesCoefficient[InverseSeries[Series[x (1 - (n - 3) x)/(1 + x)^3, {x, 0, n}], x], {x, 0, n}], {n, 1, 18}]

Crossrefs

Cf. A001764, A060354, A113207, A179848, A263843, A323208, A365816, A365817, A365818, A366204.

Sequence in context: A139383 A216351 A365863 * A130529 A075631 A127182

Adjacent sequences: A366200 A366201 A366202 * A366204 A366205 A366206

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 04 2023

Status

approved