OFFSET
0,2
COMMENTS
Binomial transform of A387358. - Seiichi Manyama, Sep 26 2025
FORMULA
a(n) = [x^n] (1 + x + x * (1 + x)^3)^n.
a(n) = Sum_{k=0..n} binomial(n,k) * A387358(k). - Seiichi Manyama, Sep 26 2025
From Vaclav Kotesovec, Oct 19 2025: (Start)
Recurrence: 3*n*(3*n - 2)*(3*n - 1)*(77220*n^3 - 398970*n^2 + 665331*n - 359485)*a(n) = 6*(2084940*n^6 - 13899600*n^5 + 35705232*n^4 - 44940249*n^3 + 28970808*n^2 - 8979979*n + 1027040)*a(n-1) + 3*(n-1)*(694980*n^5 - 4285710*n^4 + 9153999*n^3 - 8004915*n^2 + 2544894*n - 191296)*a(n-2) + 18*(n-2)*(n-1)*(386100*n^4 - 1801800*n^3 + 2635965*n^2 - 1377837*n + 212324)*a(n-3) - 23*(n-3)*(n-2)*(n-1)*(77220*n^3 - 167310*n^2 + 99051*n - 15904)*a(n-4).
a(n) ~ sqrt(4 + sqrt(3) + 12*sqrt(27/208 + sqrt(3)/13)) * (3/2 + sqrt(3) + sqrt(17 + 100/(3*sqrt(3)))/2)^n / (2*sqrt(6*Pi*n)). (End)
MATHEMATICA
Table[Sum[Binomial[n, k]*Binomial[n + 2*k, n - k], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 19 2025 *)
PROG
(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(n+2*k, n-k));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 25 2024
STATUS
approved