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, 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.
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
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 02 2005
STATUS
approved