A382616 Expansion of 1/(1 - x/(1 - x)^3)^2.
1, 2, 9, 34, 124, 444, 1567, 5466, 18885, 64732, 220403, 746166, 2513678, 8431650, 28175256, 93834240, 311565255, 1031723268, 3408137644, 11233323692, 36950587185, 121319416734, 397649266199, 1301332828086, 4252515425757, 13877722224278, 45232020345642
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..400
- Index entries for linear recurrences with constant coefficients, signature (8,-22,26,-17,6,-1).
Programs
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Magma
R
:= PowerSeriesRing(Rationals(), 40); f := 1/(1 - x/(1 - x)^3)^2; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 02 2025 -
Mathematica
Table[Sum[(k+1)*Binomial[n+2*k-1,n-k],{k,0,n}],{n,0,26}] (* Vincenzo Librandi, Apr 02 2025 *) -
PARI
a(n) = sum(k=0, n, (k+1)*binomial(n+2*k-1, n-k));
Formula
a(n) = Sum_{k=0..n} (k+1) * binomial(n+2*k-1,n-k).
a(n) = 8*a(n-1) - 22*a(n-2) + 26*a(n-3) - 17*a(n-4) + 6*a(n-5) - a(n-6) for n > 6.
G.f.: (x-1)^6/(x^3-3*x^2+4*x-1)^2.