A103370 Row sums of triangle A095801 (matrix square of the Narayana triangle A001263).
1, 3, 12, 57, 303, 1743, 10629, 67791, 448023, 3047745, 21235140, 150969195, 1091936745, 8016114681, 59616180828, 448459155063, 3407842605039, 26131449100821, 202011445055436, 1573171285950639, 12333030718989969
Offset: 1
Keywords
Examples
From _Paul D. Hanna_, Feb 01 2009: (Start) 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] +... A(x) = B(x)^3 where: B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 + x^5/2700 +...+ x^n/[n!*(n+1)!/2^n] +... (End)
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
- Jonathan M. Borwein, A short walk can be beautiful, 2015.
- J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
- J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
- Jonathan M. Borwein, Armin Straub and Christophe Vignat, Densities of short uniform random walks, Part II: Higher dimensions, Preprint, 2015.
Programs
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Mathematica
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 *) -
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]))} -
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
Formula
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
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
a(n) ~ 3^(2*n+5/2)/(4*Pi*n^3). - Vaclav Kotesovec, Oct 17 2012
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
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