A382406 Expansion of 1/(1 - x*(1 + x)^2)^3.

1, 3, 12, 37, 111, 315, 864, 2307, 6027, 15471, 39132, 97755, 241606, 591636, 1437078, 3465748, 8305161, 19788957, 46910232, 110686101, 260064912, 608684490, 1419591546, 3300027546, 7648265728, 17676484410, 40747630332, 93704299336, 214999206831, 492262973433

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (3,3,-8,-12,3,17,15,6,1).

Formula

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

a(n) = 3*a(n-1) + 3*a(n-2) - 8*a(n-3) - 12*a(n-4) + 3*a(n-5) + 17*a(n-6) + 15*a(n-7) + 6*a(n-8) + a(n-9).

G.f.: 1/(1-x-2*x^2-x^3)^3. - Vincenzo Librandi, Apr 10 2025

Mathematica

Table[Sum[Binomial[k+2, 2]*Binomial[2*k, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Apr 10 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(2*k, n-k));
(Magma) R<x> := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x*(1 + x)^2)^3; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 10 2025

Crossrefs

Cf. A002478, A362126.

Cf. A000217, A001628, A382614.

Sequence in context: A083215 A211958 A255610 * A022727 A290930 A264423

Adjacent sequences: A382403 A382404 A382405 * A382407 A382408 A382409

Keyword

nonn,easy

Author

Seiichi Manyama, Mar 31 2025

Status

approved