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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 (list; table; graph; refs; listen; history; text; internal format)
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)/(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.

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

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*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

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

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Last modified December 7 15:01 EST 2022. Contains 358667 sequences. (Running on oeis4.)