A386955 a(n) = Sum_{k=0..n} (k+1) * 2^k * binomial(2*n+1,n-k).

1, 7, 42, 235, 1262, 6594, 33780, 170475, 850230, 4200130, 20585228, 100220718, 485164988, 2337145360, 11210274408, 53567616267, 255110184486, 1211287208346, 5735765695260, 27093982041546, 127699233939684, 600650635811532, 2819989050992472, 13216897613555550

Offset

0,2

Links

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

Formula

a(n) = [x^n] 1/((1-3*x)^2 * (1-x)^n).

a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k-1,n-k).

a(n) = Sum_{k=0..n} (k+1) * 3^k * binomial(2*n-k-1,n-k).

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

a(n) ~ n * 3^(2*n) / 2^(n+1). - Vaclav Kotesovec, Aug 12 2025

Mathematica

Table[Sum[(k+1)*2^k*Binomial[2*n+1, n-k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 12 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (k+1)*2^k*binomial(2*n+1, n-k));
(Magma) [&+[(k+1) * 2^k * Binomial(2*n+1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 12 2025

Crossrefs

Cf. A088218, A258431, A384365, A386956.

Sequence in context: A279613 A162744 A324945 * A164072 A111995 A050152

Adjacent sequences: A386952 A386953 A386954 * A386956 A386957 A386958

Keyword

nonn

Author

Seiichi Manyama, Aug 11 2025

Status

approved