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A202710 Triangle read by rows. T(n, k) = coefficient of x^n in the Taylor expansion of [((1 - x - 2*x^2 - sqrt(1 - 2*x - 3*x^2))/(2*x^2))]^k.

%I A202710 #31 Oct 24 2024 05:40:18
%S A202710 1,2,1,4,4,1,9,12,6,1,21,34,24,8,1,51,94,83,40,10,1,127,258,267,164,
%T A202710 60,12,1,323,707,825,604,285,84,14,1,835,1940,2488,2084,1185,454,112,
%U A202710 16,1,2188,5337,7389,6890,4527,2106,679,144,18,1,5798,14728,21726,22120,16325,8838,3479,968,180,20,1
%N A202710 Triangle read by rows. T(n, k) = coefficient of x^n in the Taylor expansion of [((1 - x - 2*x^2 - sqrt(1 - 2*x - 3*x^2))/(2*x^2))]^k.
%C A202710 Triangle T(n,k)=
%C A202710 1. Riordan Array (1,((1-x-2*x^2-sqrt(1-2*x-3*x^2))/(2*x^2))) without first column.
%C A202710 2. Riordan Array (((1-x-2*x^2-sqrt(1-2*x-3*x^2))/(2*x)),((1-x-2*x^2-sqrt(1-2*x-3*x^2))/(2*x^2))) numbering triangle (0,0).
%C A202710 3. The leftmost column contains the Motzkin numbers A001006 without a(0).
%C A202710 The convolution triangle of the Motzkin numbers. - _Peter Luschny_, Oct 07 2022
%F A202710 T(n,k) = Sum_{i=1..k} (i*(-1)^(k-i)*binomial(k,i)*Sum_{j=floor(n/2)..n} binomial(j+i,-n+2*j)*binomial(n+i,j+i))/(n+i).
%F A202710 T(n,k) = k*Sum_{i=0..n-k} binomial(k+i,n-k-i)*binomial(n,i)/(k+i). - _Vladimir Kruchinin_, Dec 09 2016
%e A202710 Triangle begins:
%e A202710   1,
%e A202710   2, 1,
%e A202710   4, 4, 1,
%e A202710   9, 12, 6, 1,
%e A202710   21, 34, 24, 8, 1,
%e A202710   51, 94, 83, 40, 10, 1,
%e A202710   127, 258, 267, 164, 60, 12, 1
%p A202710 # Uses function PMatrix from A357368. Adds a row and a column for n, k = 0.
%p A202710 PMatrix(10, n -> simplify(hypergeom([(1-n)/2, -n/2], [2], 4))); # _Peter Luschny_, Oct 06 2022
%t A202710 T[n_, k_] := Binomial[n - 1, n - k] + k*Sum[Binomial[n, i]*Binomial[k + i, n - k - i]/(k + i), {i, 0, n - k - 1}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 06 2016, after _Vladimir Kruchinin_ *)
%o A202710 (Maxima) T(n,k):=sum((i*(-1)^(k-i)*binomial(k,i)*sum(binomial(j+i,-n+2*j)*binomial(n+i,j+i) ,j,floor(n/2),n))/(n+i),i,1,k);
%o A202710 (Maxima)
%o A202710 T(n,k):=+binomial(n-1,n-k)+k*sum((binomial(n,i)*binomial(k+i,n-k-i))/(k+i),i,0,n-k-1); /* _Vladimir Kruchinin_, Dec 06 2016*/
%Y A202710 Cf. A001006.
%K A202710 nonn,tabl
%O A202710 1,2
%A A202710 _Vladimir Kruchinin_, Dec 23 2011