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%I #43 Apr 06 2024 09:03:11
%S 1,3,12,57,303,1743,10629,67791,448023,3047745,21235140,150969195,
%T 1091936745,8016114681,59616180828,448459155063,3407842605039,
%U 26131449100821,202011445055436,1573171285950639,12333030718989969
%N Row sums of triangle A095801 (matrix square of the Narayana triangle A001263).
%H Vincenzo Librandi, <a href="/A103370/b103370.txt">Table of n, a(n) for n = 1..200</a>
%H Jonathan M. Borwein, <a href="https://carmamaths.org/resources/jon/beauty.pdf">A short walk can be beautiful</a>, 2015.
%H J. M. Borwein, <a href="https://carmamaths.org/resources/jon/OEIStalk.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
%H J. M. Borwein, <a href="/A060997/a060997.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
%H Jonathan M. Borwein, Armin Straub and Christophe Vignat, <a href="http://carmamaths.org/resources/jon/dwalks.pdf">Densities of short uniform random walks, Part II: Higher dimensions</a>, Preprint, 2015.
%F G.f. satisfies: A(x) = B(x)^3 where A(x) = Sum_{n>=0} a(n)*x^n/[n!*(n+1)!/2^n] and B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n]. - _Paul D. Hanna_, Feb 01 2009
%F Recurrence: (n+1)*(n+2)*a(n) = 2*(5*n^2-2)*a(n-1) - 9*(n-2)*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 17 2012
%F a(n) ~ 3^(2*n+5/2)/(4*Pi*n^3). - _Vaclav Kotesovec_, Oct 17 2012
%F G.f.: ((x-1)^2/(4*x*(1-9*x)^(2/3))*(-3*hypergeom([1/3, 1/3],[1],-27*x*(x-1)^2/(9*x-1)^2)+(3*x+1)^3*(9*x-1)^(-2)*hypergeom([4/3, 4/3],[2],-27*x*(x-1)^2/(9*x-1)^2)))-1+1/(2*x). - _Mark van Hoeij_, May 14 2013
%F G.f.: -(x-1)^2*hypergeom([1/3, 4/3],[2],-27*x*(x-1)^2/(9*x-1)^2)/(2*x*(1-9*x)^(2/3))-1+1/(2*x). - _Mark van Hoeij_, Nov 12 2023
%e From _Paul D. Hanna_, Feb 01 2009: (Start)
%e G.f.: A(x) = 1 + 3*x + 12*x^2/3 + 57*x^3/18 + 303*x^4/180 + 1743*x^5/2700 +...+ a(n)*x^n/[n!*(n+1)!/2^n] +...
%e A(x) = B(x)^3 where:
%e B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 + x^5/2700 +...+ x^n/[n!*(n+1)!/2^n] +... (End)
%t RecurrenceTable[{(n + 1) * (n + 2) * a[n] == 2 * (5 * n^2 - 2) * a[n - 1] - 9 * (n - 2) * (n - 1) * a[n - 2], a[1] == 1, a[2] == 3}, a, {n, 21}] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%o (PARI) {a(n)=if(n<1,0,sum(k=1,n,(matrix(n,n,m,j,binomial(m-1,j-1)*binomial(m,j-1)/j)^2)[n,k]))}
%o (PARI) {a(n)=local(B=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(B^3,n)*n!*(n+1)!/2^n} \\ _Paul D. Hanna_, Feb 01 2009
%Y Cf. A000108, A001263, A008277, A095801.
%K nonn,changed
%O 1,2
%A _Paul D. Hanna_, Feb 02 2005
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