A386957 a(n) = Sum_{k=0..n} 8^k * binomial(2*n+1,n-k).
1, 11, 114, 1163, 11806, 119646, 1211820, 12271179, 124251318, 1258065866, 12737997724, 128972535582, 1305848105836, 13221716621852, 133869898347264, 1355432788629963, 13723757247851046, 138953043155444562, 1406899565919247884, 14244858120395937738, 144229188529316725956
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..400
Programs
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Magma
[&+[8^k * Binomial(2*n+1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 14 2025
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Mathematica
Table[Sum[8^k*Binomial[2*n+1,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 14 2025 *) -
PARI
a(n) = sum(k=0, n, 8^k*binomial(2*n+1, n-k));
Formula
a(n) = [x^n] 1/((1-9*x) * (1-x)^(n+1)).
a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 9^k * binomial(2*n-k,n-k).
G.f.: 2/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7) ).
D-finite with recurrence 8*n*a(n) +(-113*n+16)*a(n-1) +162*(2*n-1)*a(n-2)=0. - R. J. Mathar, Aug 21 2025