A386834 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(4*n-k-1,n-k).
1, 8, 111, 1738, 28701, 488412, 8473387, 148994510, 2645999673, 47349481408, 852429930567, 15421507805106, 280126256513109, 5105764838932388, 93331970924544099, 1710369544783134614, 31412304686874624113, 578023658034894471048, 10654486069487503147135
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(4*n-k-1, n-k));
Formula
a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(3*n).
a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^(3*n)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(3*n+k-1,k).
Conjecture D-finite with recurrence +3*n*(16543753*n -26995933)*(3*n-1)*(3*n-2)*a(n) +(-1669899251*n^4 -26931977989*n^3 +131963667975*n^2 -188283072995*n +85757456660)*a(n-1) +2*(-61301926003*n^4 +515926265010*n^3 -1655392333929*n^2 +2311146075302*n -1165379619540)*a(n-2) -96*(39221117*n -50949760)*(4*n-9)*(2*n-5)*(4*n-7)*a(n-3)=0. - R. J. Mathar, Aug 19 2025