A084774 Coefficients of 1/sqrt(1-14*x+9*x^2); also, a(n) is the central coefficient of (1+7x+10x^2)^n.

1, 7, 69, 763, 8881, 106407, 1298949, 16065483, 200630241, 2524253767, 31947470149, 406281388443, 5187375332881, 66454791792487, 853788052488069, 10996378059281643, 141934540736139201, 1835494145265388167, 23776671158743933509, 308463567293772941883

Offset

0,2

Comments

G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.

Diagonal of rational functions 1/(1 - x - 2*y - 3*x*y), 1/(1 - x - 2*y*z - 3*x*y*z). - Gheorghe Coserea, Jul 06 2018

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Paul Barry, On the Central Coefficients of Riordan Matrices, Journal of Integer Sequences, 16 (2013), #13.5.1.

Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Vaclav Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012.

Formula

a(n) = Sum_{k=0..n} binomial(n,k)^2 * 2^k * 5^(n-k). - Paul D. Hanna, Sep 28 2012

D-finite with recurrence: n*a(n) = 7*(2*n-1)*a(n-1) - 9*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 14 2012

a(n) ~ sqrt(200 + 70*sqrt(10))*(7 + 2*sqrt(10))^n/(20*sqrt(Pi*n)) = (sqrt(2) + sqrt(5))^(2*n+1)/(2*10^(1/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

a(n) = 3^n * LegendreP(n, 7/3). - G. C. Greubel, May 31 2023

Mathematica

Table[Sum[Binomial[n, k]^2*2^k*5^(n-k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)
Table[n! SeriesCoefficient[E^(7 x) BesselI[0, 2 Sqrt[10] x], {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, May 10 2013 *)
Table[3^n*LegendreP[n, 7/3], {n, 0, 40}] (* G. C. Greubel, May 31 2023 *)

Prog

(PARI) for(n=0, 30, t=polcoeff((1+7*x+10*x^2)^n, n, x); print1(t", "))
(PARI) {a(n)=sum(k=0, n, binomial(n, k)^2*2^k*5^(n-k))} \\ Paul D. Hanna, Sep 28 2012
(GAP) List([0..20], n->Sum([0..n], k->Binomial(n, k)^2*2^k*5^(n-k))); # Muniru A Asiru, Jul 29 2018
(Magma) [3^n*Evaluate(LegendrePolynomial(n), 7/3) : n in [0..40]]; // G. C. Greubel, May 31 2023
(SageMath) [3^n*gen_legendre_P(n, 0, 7/3) for n in range(41)] # G. C. Greubel, May 31 2023

Crossrefs

Cf. A006442, A084771.

Sequence in context: A219330 A122010 A180911 * A025757 A243668 A265033

Adjacent sequences: A084771 A084772 A084773 * A084775 A084776 A084777

Keyword

nonn

Author

Paul D. Hanna, Jun 11 2003

Status

approved