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A188513 Riordan matrix (1/(x+sqrt(1-4x)),(1-sqrt(1-4x))/(2(x+sqrt(1-4x)))).

%I A188513 #14 Oct 20 2024 00:27:57
%S A188513 1,1,1,3,3,1,9,11,5,1,29,40,23,7,1,97,147,99,39,9,1,333,544,413,194,
%T A188513 59,11,1,1165,2025,1691,907,333,83,13,1,4135,7575,6842,4078,1725,524,
%U A188513 111,15,1,14845,28455,27464,17856,8453,2979,775,143,17,1,53791,107277,109631,76718,39851,15804,4797,1094,179,19,1
%N A188513 Riordan matrix (1/(x+sqrt(1-4x)),(1-sqrt(1-4x))/(2(x+sqrt(1-4x)))).
%C A188513 First column = sequence A081696
%C A188513 Row sums = sequence A101850
%F A188513 T(n,k) = [x^n] ((1-sqrt(1-4*x))/(2*(x+sqrt(1-4*x))))^k/(x+sqrt(1-4*x)).
%F A188513 T(n,k) = [x^(n-k)] (1-2*x)/((1-x)^(n+1)*(1-x-x^2)^(k+1)).
%F A188513 T(n,k) = sum(binomial(i+k,k)*binomial(2*n-i,n+k+i)*(2*k+3*i+1)/(n+k+i+1), i=0..floor((n-k)/2)).
%e A188513 Triangle begins:
%e A188513   1
%e A188513   1, 1
%e A188513   3, 3, 1
%e A188513   9, 11, 5, 1
%e A188513   29, 40, 23, 7, 1
%e A188513   97, 147, 99, 39, 9, 1
%e A188513   333, 544, 413, 194, 59, 11, 1
%e A188513   1165, 2025, 1691, 907, 333, 83, 13, 1
%e A188513   4135, 7575, 6842, 4078, 1725, 524, 111, 15, 1
%t A188513 Flatten[Table[Sum[Binomial[i+k,k]Binomial[2n-i,n+k+i](2k+3i+1)/(n+k+i+1),{i,0,Floor[(n-k)/2]}],{n,0,10},{k,0,n}]]
%o A188513 (Maxima) create_list(sum(binomial(i+k,k)*binomial(2*n-i,n+k+i)*(2*k+3*i+1)/(n+k+i+1),i,0,floor((n-k)/2)),n,0,10,k,0,n);
%Y A188513 Cf. A081696, A101850.
%K A188513 nonn,easy,tabl
%O A188513 0,4
%A A188513 _Emanuele Munarini_, Apr 02 2011