A386986 a(n) = Sum_{k=0..n} (k+1) * 8^k * binomial(2*n+2,n-k).

1, 20, 303, 4088, 51730, 628488, 7423899, 85904688, 978506478, 11008191800, 122603713078, 1354213651728, 14854030654372, 161966063719712, 1757042561230515, 18976059641899872, 204140891541240918, 2188510439907779064, 23389705325379996834, 249285017279237071440

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Formula

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

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

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

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

a(n) ~ 7 * n * 3^(4*n+2) / 2^(3*n+6). - Vaclav Kotesovec, Aug 12 2025

D-finite with recurrence 520*n*a(n) +(-8641*n-1633)*a(n-1) +486*(81*n-32)*a(n-2) +26244*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Aug 19 2025

Mathematica

Table[Sum[(k+1)* 8^k*Binomial[2*n+2, n-k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 14 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (k+1)*8^k*binomial(2*n+2, n-k));
(Magma) [&+[(k+1)*8^k * Binomial(2*n+2, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 14 2025

Crossrefs

Cf. A386957, A386958.

Cf. A000984, A002457, A383832.

Sequence in context: A001755 A361577 A016190 * A016188 A006300 A282372

Adjacent sequences: A386983 A386984 A386985 * A386987 A386988 A386989

Keyword

nonn

Author

Seiichi Manyama, Aug 12 2025

Status

approved