A391962 a(n) = Sum_{k=0..n} (k+1) * binomial(k,2*(n-k)).

1, 2, 3, 7, 17, 36, 72, 143, 282, 549, 1057, 2019, 3832, 7232, 13581, 25394, 47303, 87819, 162549, 300060, 552552, 1015259, 1861674, 3407433, 6226053, 11358407, 20691504, 37642816, 68395609, 124127202, 225025291, 407522127, 737316057, 1332795604, 2407151176, 4344040167

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (4,-6,6,-5,2,-1).

Formula

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

a(n) = 4*a(n-1) - 6*a(n-2) + 6*a(n-3) - 5*a(n-4) + 2*a(n-5) - a(n-6).

Mathematica

CoefficientList[Series[((1-x)^2+x^3)/((1-x)^2-x^3)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Dec 29 2025 *)

Prog

(PARI) my(A=1, B=1, C=A^2*B, N=2, M=40, x='x+O('x^M), X=1-A*x, Y=3); Vec(sum(k=0, N\2, C^k*binomial(N, 2*k)*X^(N-2*k)*x^(Y*k))/(X^2-C*x^Y)^N)
(Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)^2 + x^3) / ((1-x)^2 - x^3)^2); // Vincenzo Librandi, Dec 29 2025

Crossrefs

Cf. A005251, A391963.

Cf. A381421.

Sequence in context: A078721 A077007 A158498 * A267601 A155548 A191033

Adjacent sequences: A391959 A391960 A391961 * A391963 A391964 A391965

Keyword

nonn

Author

Seiichi Manyama, Dec 23 2025

Status

approved