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
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..400
Programs
-
Magma
[&+[(k+1)*8^k * Binomial(2*n+2, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 14 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 *) -
PARI
a(n) = sum(k=0, n, (k+1)*8^k*binomial(2*n+2, n-k));
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