A374605 a(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k)*binomial(3*n+2*k, n).
1, 13, 621, 40864, 3116125, 258687513, 22695228864, 2069939892096, 194303918495709, 18648446389798225, 1821631879087498621, 180513102382789033728, 18101940249015916366528, 1833572727177462316881472, 187323995560940882748187200, 19279943156312884441303524864, 1997221716775275248175573251037
Offset: 0
Examples
Factorization of a(8) thru a(10) showing divisibility by 11^3: a(8) = (3^6)*11^3*10667*18773 a(9) = (5^2)*7*(11^3)*(13^3)*3607*10103 a(10) = (11^3)*(13^4)*31*22699*68099.
Programs
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Maple
seq(add(binomial(n, k)^2*binomial(n+k, k)*binomial(3*n+2*k, n), k = 0..n), n = 0..20); # faster program for large n seq(simplify(binomial(3*n, n)*hypergeom([-n, -n, (3*n+1)/2, (3*n+2)/2], [1, 1, n+1/2], 1)), n = 0..20);
Formula
a(n) = binomial(3*n, n)*hypergeom([-n, -n, (3*n+1)/2, (3*n+2)/2], [1, 1, n+1/2], 1).
P-recursive: 16*n^3*(5616*n^4 - 30888*n^3 + 63459*n^2 - 57709*n + 19600)*(4*n - 1)^2*(4*n - 3)^2*a(n) = 36*(72783360*n^11 - 655050240*n^10 + 2595613248*n^9 - 5966404272*n^8 + 8824615470*n^7 - 8803399545*n^6 + 6034085115*n^5 - 2836309905*n^4 + 893904075*n^3 - 179376410*n^2 + 20562360*n - 1019200)*a(n-1) + 27*n*(5616*n^4 - 8424*n^3 + 4491*n^2 - 991*n + 78)*(3*n - 4)^3*(3*n - 5)^3*a(n-2) with a(0) = 1, a(1) = 13.
a(n) ~ 3^(9*n/2) * (1 + sqrt(3))^(6*n + 3) / (Pi^(3/2) * n^(3/2) * 2^(9*n + 9/2)). - Vaclav Kotesovec, Jul 22 2024
Comments