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A103370 Row sums of triangle A095801 (matrix square of the Narayana triangle A001263). 3
1, 3, 12, 57, 303, 1743, 10629, 67791, 448023, 3047745, 21235140, 150969195, 1091936745, 8016114681, 59616180828, 448459155063, 3407842605039, 26131449100821, 202011445055436, 1573171285950639, 12333030718989969 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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, Guttman 70th [Birthday] Meeting, 2015, revised May 2016.

J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttman 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.

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

EXAMPLE

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)

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 *)

PROG

(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

CROSSREFS

Cf. A000108, A001263, A008277, A095801.

Sequence in context: A276366 A243521 A151498 * A094149 A291695 A117107

Adjacent sequences:  A103367 A103368 A103369 * A103371 A103372 A103373

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 02 2005

STATUS

approved

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Last modified June 15 17:43 EDT 2019. Contains 324142 sequences. (Running on oeis4.)