A219197 Self-convolution equals A199033.

1, 2, 9, 46, 253, 1452, 8570, 51594, 315225, 1948010, 12147881, 76316508, 482392198, 3064987460, 19560379470, 125309993974, 805458510441, 5192500350906, 33561539356277, 217429403317006, 1411572472199649, 9181398851046632, 59821825063376124, 390382132833183204

Offset

0,2

Comments

Conjecture: a(n) is never congruent to 3 modulo 4.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} C(n+k+1,n-k)*C(2*n-k+1,k) = A199033(n).

G.f.: A(x) = G(x) / sqrt(1 - 2*x*G(x)^2 - 3*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 = g.f. of A001764.

G.f.: A(x) = Sum_{n>=0} A002426(n) * x^n * G(x)^(2*n+1), where A002426 are the central trinomial coefficients and G(x) = 1 + x*G(x)^3 = g.f. of A001764.

a(n) = Sum_{k=0..n} A002426(k) * C(3*n-k+1,n-k) * (2*k+1)/(3*n-k+1) for n>0, where A002426 are the central trinomial coefficients.

From Vaclav Kotesovec, Oct 05 2020: (Start)

Recurrence: 32*(n-1)*n*(2*n + 1)*(49*n^2 - 210*n + 222)*a(n) = 4*(n-1)*(10388*n^4 - 55104*n^3 + 96925*n^2 - 64446*n + 15006)*a(n-1) - 6*(22050*n^5 - 173439*n^4 + 536588*n^3 - 814340*n^2 + 604331*n - 174702)*a(n-2) - 81*(n-2)*(3*n - 7)*(3*n - 5)*(49*n^2 - 112*n + 61)*a(n-3).

a(n) ~ 3^(3*n + 7/4) / (Gamma(1/4) * n^(3/4) * 2^(2*n + 5/2)). (End)

Example

G.f.: A(x) = 1 + 2*x + 9*x^2 + 46*x^3 + 253*x^4 + 1452*x^5 +...
where A(x)^2 = 1 + 4*x + 22*x^2 + 128*x^3 + 771*x^4 + 4744*x^5 +...+ A199033(n)*x^n +...
Also, the g.f. A(x) satisfies: A(x) = G(x) * F(x*G(x)^2) where
F(x) = 1 + x + 3*x^2 + 7*x^3 + 19*x^4 + 51*x^5 + 141*x^6 +...+ A002426(n)*x^n +...
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...+ A001764(n)*x^n +...

Mathematica

A002426[n_] := Sum[Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}]; Table[Sum[A002426[k]*Binomial[3*n - k + 1, n - k]*(2*k + 1)/(3*n - k + 1), {k, 0, n}], {n, 0, 50} ] (* G. C. Greubel, Mar 06 2017 *)

Prog

(PARI) {a(n)=local(A2=sum(m=0, n, sum(k=0, m, binomial(m+k+1, m-k)*binomial(2*m-k+1, k))*x^m+x*O(x^n))); polcoeff(A2^(1/2), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(G/sqrt(1-2*x*G^2-3*x^2*G^4), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {A002426(n)=sum(k=0, n\2, binomial(n, 2*k)*binomial(2*k, k))}
{a(n)=if(n==0, 1, sum(k=0, n, A002426(k)*binomial(3*n-k+1, n-k)*(2*k+1)/(3*n-k+1)))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {A097893(n)=sum(m=0, n, sum(k=0, m\2, binomial(m, 2*k)*binomial(2*k, k)))}
{a(n)=if(n==0, 1, sum(k=0, n, A097893(k)*binomial(3*n-k, n-k)*2*k/(3*n-k)))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A199033, A183161, A002426.

Sequence in context: A268171 A168431 A036726 * A340942 A270386 A181997

Adjacent sequences: A219194 A219195 A219196 * A219198 A219199 A219200

Keyword

nonn

Author

Paul D. Hanna, Nov 14 2012

Status

approved