cp's OEIS Frontend

A121576 Riordan array (2-2*x-sqrt(1-8*x+4*x^2), (1-2*x-sqrt(1-8*x+4*x^2))/2).

%I A121576 #23 Sep 04 2025 21:55:42
%S A121576 1,2,1,6,5,1,24,24,8,1,114,123,51,11,1,600,672,312,87,14,1,3372,3858,
%T A121576 1914,618,132,17,1,19824,22992,11904,4218,1068,186,20,1,120426,140991,
%U A121576 75183,28383,8043,1689,249,23,1,749976,884112,481704,190347,58398,13929,2508,321,26,1
%N A121576 Riordan array (2-2*x-sqrt(1-8*x+4*x^2), (1-2*x-sqrt(1-8*x+4*x^2))/2).
%C A121576 Inverse of Riordan array (1/(1+2*x), x*(1-x)/(1+2*x)).
%C A121576 Row sums are A047891; first column is A054872. Signed version given by A121575.
%C A121576 Triangle T(n,k), 0 <= k <= n, read by rows, given by [2, 1, 3, 1, 3, 1, 3, 1, 3, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Aug 09 2006
%H A121576 G. C. Greubel, <a href="/A121576/b121576.txt">Rows n=0..100 of triangle, flattened</a>
%F A121576 T(n,k) = [x^(n-k)](1-2*x-2*x^2)*(1+2*x)^n/(1-x)^(n+1) = (1/2)*Sum_{i=0..n-k} binomial(n,i) * binomial(2*n-k-i,n) * (4 - 9*i + 3*i^2 - 6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i. - _Emanuele Munarini_, May 18 2011
%e A121576 Triangle begins
%e A121576      1;
%e A121576      2,    1;
%e A121576      6,    5,    1;
%e A121576     24,   24,    8,   1;
%e A121576    114,  123,   51,  11,   1;
%e A121576    600,  672,  312,  87,  14,  1;
%e A121576   3372, 3858, 1914, 618, 132, 17, 1;
%e A121576 From _Paul Barry_, Apr 27 2009: (Start)
%e A121576 Production matrix is
%e A121576   2, 1,
%e A121576   2, 3, 1,
%e A121576   2, 3, 3, 1,
%e A121576   2, 3, 3, 3, 1,
%e A121576   2, 3, 3, 3, 3, 1,
%e A121576   2, 3, 3, 3, 3, 3, 1,
%e A121576   2, 3, 3, 3, 3, 3, 3, 1
%e A121576 In general, the production matrix of the inverse of (1/(1-rx),x(1-x)/(1-rx)) is
%e A121576   -r, 1,
%e A121576   -r, 1 - r, 1,
%e A121576   -r, 1 - r, 1 - r, 1,
%e A121576   -r, 1 - r, 1 - r, 1 - r, 1,
%e A121576   -r, 1 - r, 1 - r, 1 - r, 1 - r, 1,
%e A121576   -r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1,
%e A121576   -r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1 (End)
%t A121576 Flatten[Table[Sum[Binomial[n,i]Binomial[2n-k-i,n](4-9i+3i^2-6(i-1)n+2n^2)/((n-i+2)(n-i+1))2^i,{i,0,n-k}]/2,{n,0,8},{k,0,n}]]
%t A121576 (* _Emanuele Munarini_, May 18 2011 *)
%o A121576 (Maxima) create_list(sum(binomial(n,i)*binomial(2*n-k-i,n)*(4-9*i+3*i^2-6*(i-1)*n+2*n^2)/((n-i+2)*(n-i+1))*2^i,i,0,n-k)/2,n,0,9,k,0,n); /* _Emanuele Munarini_, May 18 2011 */
%o A121576 (PARI) for(n=0,10, for(k=0,n, print1(sum(j=0, n-k, 2^j*binomial(n,j) *binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1)))/2, ", "))) \\ _G. C. Greubel_, Nov 02 2018
%o A121576 (Magma) [[(&+[ 2^j*Binomial(n,j)*Binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1))/2: j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Nov 02 2018
%K A121576 easy,nonn,tabl,changed
%O A121576 0,2
%A A121576 _Paul Barry_, Aug 08 2006