A110682 A convolution triangle of numbers based on A027307.

1, 2, 1, 10, 4, 1, 66, 24, 6, 1, 498, 172, 42, 8, 1, 4066, 1360, 326, 64, 10, 1, 34970, 11444, 2706, 536, 90, 12, 1, 312066, 100520, 23526, 4672, 810, 120, 14, 1, 2862562, 911068, 211546, 42024, 7410, 1156, 154, 16, 1

Offset: 0

Author

Philippe Deléham, Sep 15 2005

Keywords

• nonn
• tabl

Comments

Triangle T(n,k) for A(x)^k = Sum_{n>=k} T(n,k)*x^n, where o.g.f. A(x) satisfies A(x) = (1+x*A(x)^2)/(1-x*A(x)^2). - Vladimir Kruchinin, Mar 16 2011

Links

• G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
• Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Crossrefs

Columns: A027307, A032349, A033296.

Programs

Mathematica

T[n_, k_] := (k/(2*n - k))*Sum[Binomial[2*n - k, n - k - j]*Binomial[2*n - k + j - 1, 2*n - k - 1], {j, 0, n - k}]; Table[T[n, k], {n, 0, 25}, {k, 1, n}] // Flatten (* G. C. Greubel, Sep 05 2017 *)

PARI

for(n=0,25, for(k=1,n, print1((k/(2*n-k))*sum(i=0,n-k, binomial(2*n-k,n-k-i)*binomial(2*n-k+i-1,2*n-k-1)), ", "))) \\ G. C. Greubel, Sep 05 2017

Formula

T(0, 0) = 1; T(n, k) = 0 if k<0 or if k>n; T(n, k) = Sum_{j, j>=0} T(n-1, k-1+j)*A006318(j).

Sum_{k, k>=0} T(n, k) = A108442(n+1).

T(n,k) = k/(2*n-k)*Sum_{i=0,n-k} binomial(2*n-k,n-k-i)*binomial(2*n-k+i-1,2*n-k-1), n >= k > 0. - Vladimir Kruchinin, Mar 16 2011