

A110682


A convolution triangle of numbers based on A027307.


1



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
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OFFSET

0,2


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)/(1x*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], 20112013.


FORMULA

T(0, 0) = 1; T(n, k) = 0 if k<0 or if k>n; T(n, k) = Sum_{j, j>=0} T(n1, k1+j)*A006318(j).
Sum_{k, k>=0} T(n, k) = A108442(n+1).
T(n,k) = k/(2*nk)*Sum_{i=0,nk} binomial(2*nk,nki)*binomial(2*nk+i1,2*nk1), n >= k > 0.  Vladimir Kruchinin, Mar 16 2011


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 *)


PROG

(PARI) for(n=0, 25, for(k=1, n, print1((k/(2*nk))*sum(i=0, nk, binomial(2*nk, nki)*binomial(2*nk+i1, 2*nk1)), ", "))) \\ G. C. Greubel, Sep 05 2017


CROSSREFS

Columns: A027307, A032349, A033296.
Sequence in context: A142963 A099755 A202483 * A110327 A105615 A136216
Adjacent sequences: A110679 A110680 A110681 * A110683 A110684 A110685


KEYWORD

nonn,tabl


AUTHOR

Philippe Deléham, Sep 15 2005


STATUS

approved



