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

%I

%S 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 59,11,1,1165,2025,1691,907,333,83,13,1,4135,7575,6842,4078,1725,524,

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

%C Triangle begins:

%C 1

%C 1, 1

%C 3, 3, 1

%C 9, 11, 5, 1

%C 29, 40, 23, 7, 1

%C 97, 147, 99, 39, 9, 1

%C 333, 544, 413, 194, 59, 11, 1

%C 1165, 2025, 1691, 907, 333, 83, 13, 1

%C 4135, 7575, 6842, 4078, 1725, 524, 111, 15, 1

%C First column = sequence A081696

%C Row sums = sequence A101850

%F T(n,k) = [x^n] ((1-sqrt(1-4*x))/(2*(x+sqrt(1-4*x)))^k/(x+sqrt(1-4*x)).

%F T(n,k) = [x^(n-k)] (1-2*x)/((1-x)^(n+1)*(1-x-x^2)^(k+1)).

%F 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)).

%t 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 (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 Cf. A081696, A101850.

%K nonn,easy,tabl

%O 0,4

%A _Emanuele Munarini_, Apr 02 2011

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