A382614 Expansion of 1/(1 - x*(1 + x)^3)^3.

1, 3, 15, 55, 198, 681, 2263, 7341, 23331, 72928, 224814, 684882, 2065346, 6173466, 18310212, 53935350, 157904130, 459755694, 1332010954, 3841812480, 11035346151, 31579747613, 90061069065, 256028590665, 725715896698, 2051465107719, 5784472106577, 16271956316851

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Index entries for linear recurrences with constant coefficients, signature (3,6,-8,-33,-24,39,108,123,84,36,9,1).

Formula

a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(3*k,n-k).

a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3) - 33*a(n-4) - 24*a(n-5) + 39*a(n-6) + 108*a(n-7) + 123*a(n-8) + 84*a(n-9) + 36*a(n-10) + 9*a(n-11) + a(n-12).

G.f.: -1/(x^4+3*x^3+3x^2+x-1)^3. - Vincenzo Librandi, Apr 02 2025

Mathematica

Table[Sum[Binomial[k+2, 2]*Binomial[3*k, n-k], {k, 0, n}], {n, 0, 27}] (* Vincenzo Librandi, Apr 02 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(3*k, n-k));
(Magma) R<x> := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x*(1 + x)^3)^3; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 02 2025

Crossrefs

Column k=3 of A362125.

Cf. A099234, A382613.

Cf. A000217, A001628, A382406.

Cf. A382615.

Sequence in context: A286185 A152896 A007973 * A261737 A015249 A084152

Adjacent sequences: A382611 A382612 A382613 * A382615 A382616 A382617

Keyword

nonn,easy

Author

Seiichi Manyama, Mar 31 2025

Status

approved