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A084772 Coefficients of 1/sqrt(1 - 12*x + 16*x^2); also, a(n) is the central coefficient of (1 + 6*x + 5*x^2)^n. 4
1, 6, 46, 396, 3606, 33876, 324556, 3151896, 30915046, 305543556, 3038019876, 30354866856, 304523343996, 3065412858696, 30946859111256, 313206733667376, 3176825392214406, 32284147284682596, 328643023505612596 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Diagonal of rational functions 1/(1 - x - y - 4*x*y), 1/(1 - x - y*z - 4*x*y*z). - Gheorghe Coserea, Jul 06 2018

LINKS

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

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.

H. A. Verrill, Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations, arXiv:math/0407327 [math.CO], 2008, Theorem 8.

FORMULA

a(n) = Sum_{k=0..n} 5^k*C(n,k)^2. - Benoit Cloitre, Oct 26 2003

E.g.f.: exp(6*x)*BesselI(0, 2*sqrt(5)*x). - Paul Barry, Sep 20 2004

Asymptotic: a(n) ~ (1+sqrt(5))^(2*n+1)/(2*5^(1/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Sep 11 2012

D-finite with recurrence: n*a(n) + 6*(1-2*n)*a(n-1) + 16*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 09 2012

EXAMPLE

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.

MATHEMATICA

Table[n! SeriesCoefficient[E^(6 x) BesselI[0, 2 Sqrt[5] x], {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, May 10 2013 *)

CoefficientList[Series[1/Sqrt[1-12x+16x^2], {x, 0, 30}], x] (* Harvey P. Dale, Apr 17 2015 *)

PROG

(PARI) for(n=0, 30, t=polcoeff((1+6*x+5*x^2)^n, n, x); print1(t", "))

(GAP) List([0..20], n->Sum([0..n], k->Binomial(n, k)^2*5^k)); # Muniru A Asiru, Jul 29 2018

CROSSREFS

Cf. A001850.

Sequence in context: A301421 A288689 A271933 * A199563 A349332 A111531

Adjacent sequences: A084769 A084770 A084771 * A084773 A084774 A084775

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 10 2003

STATUS

approved

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Last modified February 4 19:58 EST 2023. Contains 360059 sequences. (Running on oeis4.)