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%I #17 May 30 2022 06:06:19
%S 1,2,1,4,4,1,9,12,6,1,23,34,24,8,1,65,98,83,40,10,1,197,294,273,164,
%T 60,12,1,626,919,891,612,285,84,14,1,2056,2974,2938,2188,1195,454,112,
%U 16,1,6918,9891,9846,7698,4677,2118,679,144,18,1,23714,33604,33549
%N Triangle T(n,k) = coefficient of x^n in expansion of ((1-sqrt(1-4*x))/((1-x)*2))^k = sum(n>=k, T(n,k) * x^n).
%C Triangle T(n,k)=
%C 1. Riordan Array (1,(1-sqrt(1-4*x))/((1-x)*2)) without first column.
%C 2. Riordan Array ((1-sqrt(1-4*x))/((1-x)*2*x),(1-sqrt(1-4*x))/((1-x)*2)) numbering triangle (0,0).
%C Convolution triangle of A014137(n). - _Philippe Deléham_, Jan 23 2014
%F T(n,k):=k*sum(i=0..n-k, (binomial(i+k-1,k-1)*binomial(2*(n-i)-k-1,n-i-1))/(n-i)).
%e Triangle:
%e 1,
%e 2, 1,
%e 4, 4, 1,
%e 9, 12, 6, 1,
%e 23, 34, 24, 8, 1,
%e 65, 98, 83, 40, 10, 1,
%e 197, 294, 273, 164, 60, 12, 1
%t T[n_, k_]:= (k/n) (Binomial[-1 - k + 2 n, -1 + n] HypergeometricPFQ[{k, k - n, -n}, {1/2 + k/2 - n, 1 + k/2 - n}, 1/4]);
%t Table[T[n, k], {n, 1, 9}, {k, 1, n}] // TableForm (* _Peter Luschny_, May 30 2022 *)
%o (Maxima)
%o T(n,k):=k*sum((binomial(i+k-1,k-1)*binomial(2*(n-i)-k-1,n-i-1))/(n-i),i,0,n-k);
%Y Cf. Columns: A014137, A014143
%K nonn,tabl
%O 1,2
%A _Vladimir Kruchinin_, Nov 25 2011
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