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A103370
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Row sums of triangle A095801 (matrix square of the Narayana triangle A001263).
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3
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1, 3, 12, 57, 303, 1743, 10629, 67791, 448023, 3047745, 21235140, 150969195, 1091936745, 8016114681, 59616180828, 448459155063, 3407842605039, 26131449100821, 202011445055436, 1573171285950639, 12333030718989969
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OFFSET
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1,2
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LINKS
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FORMULA
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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
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
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EXAMPLE
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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)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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