OFFSET
0,2
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.
Rob Noble, Asymptotics of a family of binomial sums, J. Number Theory 130 (2010), no. 11, 2561-2585.
FORMULA
G.f.: 1/sqrt(1-14*x+25*x^2).
E.g.f.: exp(7*x)*BesselI(0, 2*sqrt(6)*x).
a(n) = Sum_{k=0..n} C(n, k)^2*6^k.
a(n) = [x^n] (1+7*x+6*x^2)^n.
From Vaclav Kotesovec, Sep 15 2012: (Start)
General recurrence for Sum_{k=0..n} C(n,k)^2*x^k (this is case x=6): (n+2)*a(n+2)-(x+1)*(2*n+3)*a(n+1)+(x-1)^2*(n+1)*a(n)=0.
Asymptotic (Rob Noble, 2010): a(n) ~ (1+sqrt(x))^(2*n+1)/(2*x^(1/4)*sqrt(Pi*n)), this is case x=6. (End)
D-finite with recurrence: n*a(n) +7*(-2*n+1)*a(n-1) +25*(n-1)*a(n-2)=0. - R. J. Mathar, Jan 20 2020
a(n) = 5^n*hypergeom([-n, n + 1], [1], -1/5). - Detlef Meya, May 24 2024
MATHEMATICA
Table[Sum[Binomial[n, k]^2*6^k, {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Sep 15 2012 *)
CoefficientList[Series[1/Sqrt[(1-7*x)^2-24*x^2], {x, 0, 25}], x] (* Stefano Spezia, Dec 04 2018 *)
a[n_] := 5^n*HypergeometricPFQ[{-n, n+1}, {1}, -1/5]; Table[a[n], {n, 0, 19}] (* Detlef Meya, May 24 2024 *)
PROG
(PARI) x='x+O('x^66); Vec(1/sqrt(1-14*x+25*x^2)) \\ Joerg Arndt, May 12 2013
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 20 2004
STATUS
approved