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

0,2

Comments

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

Links

Table of n, a(n) for n=0..14.

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)

Example

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)

Prog

(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

Crossrefs

Cf. A001263, A105556, A155926.

Sequence in context: A228551 A001927 A289987 * A322399 A340026 A279011

Adjacent sequences: A105555 A105556 A105557 * A105559 A105560 A105561

Keyword

nonn

Author

Paul D. Hanna, Apr 14 2005

Status

approved