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
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).
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[3*k,n-k],{k,0,n}],{n,0,27}] (* Vincenzo Librandi, Apr 02 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(3*k, n-k));
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