A180717 G.f.: Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2*x^k]^2 * x^n.

1, 1, 3, 10, 37, 140, 544, 2181, 8873, 36647, 152950, 644313, 2734648, 11681428, 50173541, 216532005, 938383331, 4081653710, 17811999929, 77957939080, 342099306436, 1504801777973, 6633574235109, 29300516237855

Offset

0,3

Comments

Compare g.f. to a g.f. of the Whitney numbers (A051286):

. Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2*x^k] * x^n.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..900

Formula

a(n) ~ c * d^n / (Pi*n), where d = 1/3*(5 + (187/2 - (9*sqrt(93))/2)^(1/3) + (1/2*(187 + 9*sqrt(93)))^(1/3)) = 4.61347026758155538... is the root of the equation 1 - 2*d + 5*d^2 - d^3 = 0, c = 1/192*(80 + (382976 - 18432*sqrt(93))^(1/3) + 8*2^(2/3)*(187 + 9*sqrt(93))^(1/3)) = 1.15336756689... is the root of the equation 64*c^3 - 80*c^2 + 8*c - 1 = 0. - Vaclav Kotesovec, Jul 31 2014

Example

G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 140*x^5 + 544*x^6 +...
equals the sum of the series:
A(x) = 1 + (1 + x)^2*x + (1 + 4*x + x^2)^2*x^2
+ (1 + 9*x + 9*x^2 + x^3)^2*x^3
+ (1 + 16*x + 36*x^2 + 16*x^3 + x^4)^2*x^4
+ (1 + 25*x + 100*x^2 + 100*x^3 + 25*x^4 + x^5)^2*x^5
+ (1 + 36*x + 225*x^2 + 400*x^3 + 225*x^4 + 36*x^5 + x^6)^2*x^6 +...

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=0, m, binomial(m, k)^2*x^k)^2*x^m)+x*O(x^n), n)}

Crossrefs

Cf. A180718.

Sequence in context: A242725 A151315 A164048 * A151049 A195350 A289810

Adjacent sequences: A180714 A180715 A180716 * A180718 A180719 A180720

Keyword

nonn

Author

Paul D. Hanna, Sep 29 2010

Status

approved