A262737 O.g.f. exp( Sum_{n >= 1} A262732(n)*x^n/n ).

1, 8, 95, 1336, 20642, 338640, 5791291, 102108760, 1842857390, 33879118384, 632210693270, 11944142806064, 228010741228740, 4391334026631072, 85221618348230355, 1664901954576830360, 32716286416687895862, 646228961799752926320, 12823701194384778672322

Offset: 0

Author

Peter Bala, Sep 29 2015

Keywords

• nonn
• easy

Comments

O.g.f. is 1/x * the series reversion of x*(1 - x)^k/(1 + x)^(k+2) for k = 3. See the cross references for related sequences obtained from other values of k.

Links

• Peter Bala, Notes on logarithmic differentiation, the binomial transform and series reversion

Crossrefs

Cf. A000108 (k = 0), A007297 (k = 1), A066357 (k = 2), A262738 (k = 4), A262739 (k = 5), A262740 (k = 6), A262732.

Programs

Maple

A262737 := proc (n) option remember; if n = 0 then 1 else add(1/k!*(5*k)!/GAMMA(5*k/2 + 1)*GAMMA(3*k/2 + 1)/(3*k)!*A262737(n-k), k = 1 .. n)/n end if; end proc:
seq(A262737(n), n = 0 .. 20);

PARI

a(n) = sum(k=0, n, binomial(5*(n+1),k)*binomial(4*(n+1)-k-2,(n+1)-k-1))/(n+1); \\ Altug Alkan, Oct 03 2015

Formula

a(n-1) = 1/n * Sum_{i = 0..n-1} binomial(5*n,i)*binomial(4*n-i-2,n-i-1).

O.g.f.: A(x) = exp ( Sum_{n >= 1} 1/n! * (5*n)!/(5*n/2)! * (3*n/2)!/(3*n)!*x^n/n ) = 1 + 8*x + 195*x^2 + 1336*x^3 + ....

1 + x*A'(x)/A(x) is the o.g.f. for A262732.

O.g.f. is the series reversion of x*(1 - x)^3/(1 + x)^5.

a(0) = 1 and for n >= 1, a(n) = 1/n * Sum {k = 1..n} 1/k!*(5*k)!/GAMMA(5*k/2+1)*GAMMA(3*k/2+1)/(3*k)! * a(n-k).