A105558 Central terms in even-indexed rows of triangle A105556 and thus equals the n-th row sum of the n-th matrix power of the Narayana triangle A001263.

1, 2, 12, 148, 3105, 99156, 4481449, 272312216, 21414443481, 2116193061340, 256712977920256, 37506637787774112, 6496315164318118165, 1316230822119433518312, 308426950979497974254310

Offset: 0

Author

Paul D. Hanna, Apr 14 2005

Keywords

• nonn

Comments

Each term a(n) is divisible by (n+1) for all n>=0.

Examples

			Terms a(n) divided by (n+1) begin:
1,1,4,37,621,16526,640207,34039027,2379382609,211619306134,...
Contribution from _Paul D. Hanna_, Jan 31 2009: (Start)
G.f.: A(x) = 1 + 2*x + 12*x^2/3 + 148*x^3/18 + 3105*x^4/180 +...+ a(n)*x^n/[n!*(n+1)!/2^n] +...
G.f.: A(x) = d/dx x*F(x) where F(x) = B(x*F(x)) and:
F(x) = 1 + x + 4*x^2/3 + 37*x^3/18 + 621*x^4/180 + 16526*x^5/2700 +...+ A155926(n)*x^n/[n!*(n+1)!/2^n] +...
B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 +...+ x^n/[n!*(n+1)!/2^n] +... (End)
		

Crossrefs

Cf. A001263, A105556, A155926.

Programs

PARI

a(n)=local(N=matrix(n+1,n+1,m,j,if(m>=j, binomial(m-1,j-1)*binomial(m,j-1)/j))); sum(j=0,n,(N^n)[n+1,j+1])
for(n=0,20,print1(a(n),", "))

PARI

a(n)=local(F=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(deriv(serreverse(x/F)),n)*n!*(n+1)!/2^n
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Jan 31 2009

Formula

Contribution from Paul D. Hanna, Jan 31 2009: (Start)

a(n) = (n+1)*A155926(n) for n>=0.

G.f.: A(x) = d/dx x*F(x) where F(x) = B(x*F(x)) and F(x) = Sum_{n>=0} A155926(n)*x^n/[n!*(n+1)!/2^n] with B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n] and A(x) = Sum_{n>=0} a(n)*x^n/[n!*(n+1)!/2^n]. (End)