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A156894 a(n) = Sum_{k=0..n} C(n,k)*C(2n+k-1,k). 4
1, 3, 19, 138, 1059, 8378, 67582, 552576, 4563235, 37972290, 317894394, 2674398268, 22590697614, 191475925332, 1627653567916, 13870754053388, 118464647799075, 1013709715774130, 8689197042438274, 74594573994750972, 641252293546113434, 5519339268476249676, 47558930664216470628 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

FORMULA

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

a(n) = (4*(n+1)*(2*n+1)*A003169(n+1)-(5*n+1)*(2*n-1)*A003169(n))/(17*n + 5) for n>0. - Mark van Hoeij, Jul 14 2010

a(n) = Hyper2F1([-n, 2*n], [1], -1). - Peter Luschny, Aug 02 2014

Conjecture: 64*n*(2*n-1)*a(n) +16*(-89*n^2+134*n-63)*a(n-1) +4*(661*n^2-2619*n+2576)*a(n-2) +3*(-119*n^2+713*n-1092)*a(n-3) +6*(2*n-7)*(n-4)*a(n-4)=0. - R. J. Mathar, Feb 05 2015

Conjecture: 16*n*(782*n+5365)*(2*n-1)*a(n) +8*(3128*n^3-362053*n^2+593930*n-290328)*a(n-1) +3*(-726869*n^3+5105981*n^2-11667946*n+8715544)*a(n-2) +158*(2*n-5)*(n-3)*(391*n-764)*a(n-3)=0. - R. J. Mathar, Feb 05 2015

Conjecture: 4*n*(2*n-1)*(17*n^2-52*n+39)*a(n) +(-1207*n^4+4899*n^3-6692*n^2+3504*n-576)*a(n-1) +2*(n-2)*(2*n-3)*(17*n^2-18*n+4)*a(n-2)=0. - R. J. Mathar, Feb 05 2015 [the Maple command sumrecursion (binomial(n,k)*binomial(2n+k-1,k),k,a(n)) verifies this recurrence. - Peter Bala, Oct 05 2015 ]

a(n) ~ sqrt(578 + 306*sqrt(17)) * (71 + 17*sqrt(17))^n / (17 * sqrt(Pi*n) * 2^(4*n+2)). - Vaclav Kotesovec, Feb 05 2015

exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 3*x + 14*x^2 + 79*x^3 + ... is the o.g.f. of A003169 (taken with offset 0). - Peter Bala, Oct 05 2015

MAPLE

a := n -> hypergeom([-n, 2*n], [1], -1);

seq(round(evalf(a(n), 32)), n=0..19); # Peter Luschny, Aug 02 2014

MATHEMATICA

Table[Sum[Binomial[n, k]Binomial[2n+k-1, k], {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Nov 12 2014 *)

PROG

(PARI) a(n) = if (n < 1, 1, sum(k=0, n, binomial(n, k)*binomial(2*n+k-1, k)));

vector(50, n, a(n-1)) \\ Altug Alkan, Oct 05 2015

CROSSREFS

Cf. A123164, A114496, A003169.

Sequence in context: A094662 A321349 A115750 * A221374 A073515 A074559

Adjacent sequences:  A156891 A156892 A156893 * A156895 A156896 A156897

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Feb 17 2009

STATUS

approved

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Last modified April 22 08:25 EDT 2019. Contains 322329 sequences. (Running on oeis4.)