A382615 Expansion of 1/(1 - x/(1 - x)^3)^3.
1, 3, 15, 64, 261, 1032, 3982, 15066, 56094, 206068, 748452, 2691966, 9600233, 33982197, 119495229, 417724302, 1452550371, 5026878774, 17321417650, 59450099958, 203306331429, 692955932103, 2354664287943, 7978488379398, 26963061909228, 90897971951727
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..400
- Index entries for linear recurrences with constant coefficients, signature (12,-57,139,-195,174,-102,39,-9,1).
Programs
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Magma
R
:= 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 -
Mathematica
Table[Sum[Binomial[k+2,2]*Binomial[n+2*k-1,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Apr 02 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(n+2*k-1, n-k));
Formula
a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(n+2*k-1,n-k).
a(n) = 12*a(n-1) - 57*a(n-2) + 139*a(n-3) - 195*a(n-4) + 174*a(n-5) - 102*a(n-6) + 39*a(n-7) - 9*a(n-8) + a(n-9) for n > 9.