A386956 a(n) = Sum_{k=0..n} (k+1) * 8^k * binomial(2*n+1,n-k).

1, 19, 282, 3763, 47294, 571950, 6733668, 77723187, 883589238, 9924844474, 110396411372, 1218075749934, 13348677037868, 145438914042172, 1576690043132376, 17018212213758771, 182983432175308710, 1960781840268630786, 20947171352106580284, 223169444039365834362

Offset

0,2

Links

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

Formula

a(n) = [x^n] 1/((1-9*x)^2 * (1-x)^n).

a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k-1,n-k).

a(n) = Sum_{k=0..n} (k+1) * 9^k * binomial(2*n-k-1,n-k).

G.f.: 2 * (1+sqrt(1-4*x))/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7)^2 ).

Mathematica

Table[Sum[(k+1) * 8^k*Binomial[2*n+1, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 03 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (k+1)*8^k*binomial(2*n+1, n-k));
(Magma) [&+[(k+1) * 8^k * Binomial(2*n+1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025

Crossrefs

Cf. A088218, A258431, A384365, A386955.

Cf. A386957.

Sequence in context: A328916 A081045 A155017 * A199245 A152591 A051562

Adjacent sequences: A386953 A386954 A386955 * A386957 A386958 A386959

Keyword

nonn

Author

Seiichi Manyama, Aug 11 2025

Status

approved