A391035 Expansion of g/(1 - x^4*g^4), where g = 1+x*g^2 is the g.f. of A000108.

1, 1, 2, 5, 15, 47, 152, 504, 1706, 5872, 20490, 72319, 257723, 926071, 3351552, 12205864, 44698318, 164492726, 608009172, 2256266234, 8402790998, 31395570766, 117652696392, 442095949780, 1665396815980, 6288134065462, 23793282441962, 90208829415539, 342646535562835

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..floor(n/4)} (4*k+1) * binomial(2*n-4*k+1,n-4*k)/(2*n-4*k+1).

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (4*k+1) * binomial(2*n-4*k,n-4*k).

Mathematica

Table[Sum[(4*k+1)*Binomial[2*n-4*k+1, n-4*k]/(2*n-4*k+1), {k, 0, Floor[n/4]}], {n, 0, 30}] (* Vincenzo Librandi, Dec 01 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\4, (4*k+1)*binomial(2*n-4*k+1, n-4*k)/(2*n-4*k+1));
(Magma) [&+[(4*k+1)*Binomial(2*n-4*k+1, n-4*k)/(2*n-4*k+1): k in [0..Floor(n/4)]] : n in [0..40] ]; // Vincenzo Librandi, Dec 01 2025

Crossrefs

Cf. A390479, A391030, A391032, A391034.

Cf. A000108.

Sequence in context: A151280 A149914 A071735 * A148363 A365267 A350434

Adjacent sequences: A391032 A391033 A391034 * A391036 A391037 A391038

Keyword

nonn

Author

Seiichi Manyama, Nov 26 2025

Status

approved