A391964 a(n) = Sum_{k=0..n} (k+1) * 2^k * binomial(k,2*(n-k)).

1, 4, 12, 44, 176, 672, 2448, 8704, 30528, 105920, 364032, 1241088, 4202752, 14150656, 47410176, 158157824, 525602816, 1740840960, 5748461568, 18930761728, 62190206976, 203850235904, 666838499328, 2177331363840, 7097215418368, 23097693700096, 75061550383104

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,-24,40,-48,32,-16).

Formula

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

a(n) = 8*a(n-1) - 24*a(n-2) + 40*a(n-3) - 48*a(n-4) + 32*a(n-5) - 16*a(n-6).

Mathematica

CoefficientList[Series[((1-2*x)^2+4*x^3)/((1-2*x)^2-4*x^3)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2025 *)

Prog

(PARI) my(A=2, B=1, C=A^2*B, N=2, M=30, 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:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x)^2 + 4*x^3) / ((1-2*x)^2 - 4*x^3)^2); // Vincenzo Librandi, Dec 30 2025

Crossrefs

Cf. A390700.

Sequence in context: A076793 A007860 A226855 * A393002 A039740 A065143

Adjacent sequences: A391961 A391962 A391963 * A391965 A391966 A391967

Keyword

nonn

Author

Seiichi Manyama, Dec 23 2025

Status

approved