A392076 a(n) = Sum_{k=0..n} (k+1) * binomial(k,4*(n-k)).

1, 2, 3, 4, 5, 11, 37, 113, 289, 640, 1280, 2412, 4468, 8429, 16460, 33062, 67084, 135283, 268929, 526680, 1020344, 1966429, 3787730, 7309799, 14135460, 27353769, 52882135, 102017609, 196309933, 376896805, 722393936, 1383122752, 2646430148, 5061050284

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,58,-36,20,-9,2,-1).

Formula

G.f.: (1-x)^2 * ((1-x)^4 + 3*x^5) / ((1-x)^4 - x^5)^2.

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 58*a(n-5) - 36*a(n-6) + 20*a(n-7) - 9*a(n-8) + 2*a(n-9) - a(n-10).

Mathematica

CoefficientList[Series[(1-x)^2*((1-x)^4+3*x^5)/((1-x)^4-x^5)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 31 2025 *)

Prog

(PARI) a178619(n, k) = sum(j=0, k, (-1)^(k-j)*binomial(n+1, k-j)*binomial(n+4*j, 4*j));
my(A=1, B=1, C=A^4*B, N=2, M=40, x='x+O('x^M), X=1-A*x, Y=5); Vec(sum(k=0, (3*N)\4, C^k*a178619(N-1, k)*X^(3*N-4*k)*x^(Y*k))/(X^4-C*x^Y)^N)
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1-x)^2 * ((1-x)^4 + 3*x^5) / ((1-x)^4 - x^5)^2); // Vincenzo Librandi, Dec 31 2025

Crossrefs

Cf. A001629, A391962, A392044.

Cf. A005676, A178619.

Sequence in context: A032988 A280206 A190783 * A136367 A014545 A379046

Adjacent sequences: A392073 A392074 A392075 * A392077 A392078 A392079

Keyword

nonn

Author

Seiichi Manyama, Dec 30 2025

Status

approved