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A391891
a(n) = Sum_{k=0..n} (k+1) * binomial(2*k+1,2*n-2*k).
2
1, 2, 9, 34, 104, 326, 995, 2954, 8671, 25156, 72256, 205948, 583109, 1641570, 4598725, 12827750, 35646792, 98726370, 272611239, 750730314, 2062369787, 5653126280, 15464460608, 42225863096, 115102499721, 313264141570, 851348070593, 2310572001450, 6263085581800
OFFSET
0,2
FORMULA
G.f.: ((1-x-x^2)^2 + 4*x^2 - 4*x^4) / ((1-x-x^2)^2 - 4*x^3)^2.
a(n) = 4*a(n-1) - 2*a(n-2) - 11*a(n-4) - 2*a(n-6) + 4*a(n-7) - a(n-8).
MATHEMATICA
CoefficientList[Series[((1-x-x^2)^2+4*x^2-4*x^4)/((1-x-x^2)^2-4*x^3)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Jan 01 2026 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(((1-x-x^2)^2+4*x^2-4*x^4)/((1-x-x^2)^2-4*x^3)^2)
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x-x^2)^2 + 4*x^2 - 4*x^4) / ((1-x-x^2)^2 - 4*x^3)^2); // Vincenzo Librandi, Jan 01 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Dec 22 2025
STATUS
approved