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%I #17 May 11 2022 13:01:19
%S 1,2,7,31,142,659,3113,14918,72199,351983,1726022,8504509,42070429,
%T 208812722,1039387519,5186451311,25935769702,129942777227,
%U 652133298421,3277734587302,16496741964221,83129076840317,419362231888882
%N a(n) = Sum_{k=0..n} binomial(n,k)^2*F(k+1).
%H Alois P. Heinz, <a href="/A114198/b114198.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>.
%F a(n) = Sum_{k=0..n} C(n, k)^2 * F(k+1); a(n) = A114197(2n, n).
%F a(n) = (phi^(n-1) * P_n(sqrt(5)-2) - (1-phi)^(n-1) * P_n(-sqrt(5)-2))/sqrt(5), where phi = (1+sqrt(5))/2, P_n(x) is the Legendre polynomial.
%F a(n) ~ sqrt((6 + 2*sqrt(5) + sqrt(2*(29 + 13*sqrt(5))))/10)/2 * ((3 + sqrt(5))/2 + sqrt(2*(1+sqrt(5))))^n / sqrt(Pi*n). - _Vaclav Kotesovec_, May 06 2017
%F a(n) ~ sqrt(2*phi^2 + phi^(7/2)) * (2*phi^(1/2) + phi^2)^n / (2*sqrt(5*Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Sep 22 2017
%F D-finite with recurrence +n*(n-1)*a(n) -5*n*(n-1)*a(n-1) +2*(-n^2+17*n-27)*a(n-2) +(11*n^2-135*n+270)*a(n-3) +2*(-17*n^2+121*n-215)*a(n-4) +(n-4)*(43*n-191)*a(n-5) -3*(n-4)*(n-5)*a(n-6)=0. - _R. J. Mathar_, May 11 2022
%p a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 31][n+1],
%p ((3*(n-1))*(2*n-5)*(13*n^2-26*n+10) *a(n-1)
%p -(7*n^2-14*n+6)*(13*n^2-52*n+49) *a(n-2)
%p +(n-2)*(182*n^3-819*n^2+1050*n-351) *a(n-3)
%p -(n-2)*(n-3)*(13*n^2-26*n+10) *a(n-4))/
%p (n*(n-1)*(13*n^2-52*n+49)))
%p end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Sep 28 2016
%t FullSimplify@Table[(GoldenRatio^(n - 1) LegendreP[n, Sqrt[5] - 2] - (1 - GoldenRatio)^(n - 1) LegendreP[n, -Sqrt[5] - 2])/Sqrt[5], {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 28 2016 *)
%Y Cf. A000045, A219672, A219673.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Nov 16 2005