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A183161 Self-convolution equals A183160. 5
1, 1, 5, 26, 145, 841, 5006, 30350, 186537, 1158685, 7258145, 45779420, 290399030, 1851032314, 11847434810, 76100034106, 490343021881, 3168174174105, 20520045125681, 133197288251330, 866293102078525, 5644234561103785 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Conjecture: a(n) is never congruent to 3 modulo 4; see A218622. - Paul D. Hanna, Nov 03 2012

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} C(n+k,n-k)*C(2*n-k,k) = A183160(n).

G.f.: A(x) = 1/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. - Paul D. Hanna, Nov 03 2012

G.f.: A(x) = Sum_{n>=0} A002426(n) * x^n * G(x)^(2*n), where A002426 are the central trinomial coefficients and G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 03 2012

a(n) = Sum_{k=0..n} A002426(k) * C(3*n-k,n-k) * 2*k/(3*n-k) for n>0, where A002426 are the central trinomial coefficients: A002426(n) = Sum_{k=0..[n/2]} C(n,2*k)*C(2*k,k). - Paul D. Hanna, Nov 04 2012

G.f.: A(x) = 1/sqrt(1 + 3*x*G(x) - 5*x*G(x)^2), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Jun 16 2013

EXAMPLE

G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 145*x^4 + 841*x^5 + 5006*x^6 +...

A(x)^2 = 1 + 2*x + 11*x^2 + 62*x^3 + 367*x^4 + 2232*x^5 + 13820*x^6 + 86662*x^7 +...+ A183160(n)*x^n +...

PROG

(PARI) a(n)=local(A2=sum(m=0, n, sum(k=0, m, binomial(m+k, m-k)*binomial(2*m-k, k))*x^m+x*O(x^n))); polcoeff(A2^(1/2), n)

(PARI) a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/sqrt(1-2*x*G^2-3*x^2*G^4), n) \\ Paul D. Hanna, Nov 03 2012

(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, n-k)*2*k/(3*n-k)))} \\ Paul D. Hanna, Nov 04 2012

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(1/sqrt(1+3*x*G-5*x*G^2), n)

for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 16 2013

CROSSREFS

Cf. A183160, A218622, A001764, A002426, A219197.

Sequence in context: A045379 A053487 A277957 * A263134 A082029 A081047

Adjacent sequences:  A183158 A183159 A183160 * A183162 A183163 A183164

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 27 2010

STATUS

approved

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Last modified January 19 18:25 EST 2020. Contains 331051 sequences. (Running on oeis4.)