A388533 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+2*k,k).

1, 4, 24, 163, 1168, 8624, 64911, 495156, 3814624, 29610964, 231238624, 1814615136, 14297530351, 113034862828, 896235865528, 7123915585883, 56749595980352, 452939757433792, 3621239489720964, 28995718924420464, 232489398617644608, 1866412543947072312

Offset

0,2

Comments

Binomial transform of A370286.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A370286(k).

a(n) = [x^n] ((1+x)^3 + x + x^2)^n.

From Peter Bala, Sep 23 2025: (Start)

The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.

Equivalently, exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 4*x + 20*x^2 + 113*x^3 + 688*x^4 + 4404*x^5 + ... has integer coefficients. See A108447.

2*(2*n - 1)*(74*n - 95)*n*a(n) = 8*(296*n^3 - 676*n^2 + 440*n - 81)*a(n-1) + 16*(n - 1)*(37*n^2 - 66*n + 12)*a(n-2) - 5*(n - 1)*(n - 2)*(74*n - 21)*a(n-3) with a(0) = 1, a(1) = 4 and a(2) = 24. (End)

a(n) ~ sqrt(1/4 + 2*sqrt(10/111)*cos(arccos(3*sqrt(111/10)/10)/3)) * (8/3 + 2*sqrt(70)*cos(arccos(4537/(560*sqrt(70)))/3)/3)^n / sqrt(Pi*n). - Vaclav Kotesovec, Oct 19 2025

Mathematica

Table[Sum[Binomial[n, k]*Binomial[n+2*k, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Sep 19 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(n+2*k, k));
(Magma) [&+[Binomial(n, k)*Binomial(n+2*k, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 19 2025

Crossrefs

Cf. A001850, A388534, A388535, A388536.

Cf. A108447, A370286, A387105.

Sequence in context: A343842 A027079 A213441 * A238299 A221656 A366623

Adjacent sequences: A388530 A388531 A388532 * A388534 A388535 A388536

Keyword

nonn

Author

Seiichi Manyama, Sep 18 2025

Status

approved