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A155761 Riordan array (c(2*x^2), x*c(2*x^2)) where c(x) is the g.f. of A000108.

%I A155761 #11 Jun 07 2021 04:38:26
%S A155761 1,0,1,2,0,1,0,4,0,1,8,0,6,0,1,0,20,0,8,0,1,40,0,36,0,10,0,1,0,112,0,
%T A155761 56,0,12,0,1,224,0,224,0,80,0,14,0,1,0,672,0,384,0,108,0,16,0,1,1344,
%U A155761 0,1440,0,600,0,140,0,18,0,1
%N A155761 Riordan array (c(2*x^2), x*c(2*x^2)) where c(x) is the g.f. of A000108.
%C A155761 Inverse of Riordan array (1/(1+2*x^2), x/(1+2*x^2)).
%H A155761 G. C. Greubel, <a href="/A155761/b155761.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A155761 T(n,k) = (1+(-1)^(n-k)) * ((k+1)/(n+1)) * binomial(n+1, (n-k)/2) * 2^((n-k-2)/2).
%F A155761 Sum_{k=0..n} T(n, k) = A126087(n).
%F A155761 T(n,k) = 2^((n-k)/2) * A053121(n,k). - _Philippe Deléham_, Feb 11 2009
%F A155761 Sum_{k=0..n} T(2*n-k, k) = A064062(n+1). - _G. C. Greubel_, Jun 06 2021
%e A155761 Triangle begins:
%e A155761     1;
%e A155761     0,   1;
%e A155761     2,   0,   1;
%e A155761     0,   4,   0,  1;
%e A155761     8,   0,   6,  0,  1;
%e A155761     0,  20,   0,  8,  0,  1;
%e A155761    40,   0,  36,  0, 10,  0,  1;
%e A155761     0, 112,   0, 56,  0, 12,  0, 1;
%e A155761   224,   0, 224,  0, 80,  0, 14, 0, 1;
%e A155761 Production matrix begins as:
%e A155761   0, 1;
%e A155761   2, 0, 1;
%e A155761   0, 2, 0, 1;
%e A155761   0, 0, 2, 0, 1;
%e A155761   0, 0, 0, 2, 0, 1;
%e A155761   0, 0, 0, 0, 2, 0, 1;
%e A155761   0, 0, 0, 0, 0, 2, 0, 1;
%e A155761   0, 0, 0, 0, 0, 0, 2, 0, 1;
%e A155761   0, 0, 0, 0, 0, 0, 0, 2, 0, 1;
%e A155761   0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1;
%t A155761 T[n_, k_]:= (1+(-1)^(n-k))*2^((n-k-2)/2)*((k+1)/(n+1))*Binomial[n+1, (n-k)/2];
%t A155761 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 06 2021 *)
%o A155761 (Sage)
%o A155761 def A155761(n,k): return (1+(-1)^(n-k))*2^((n-k-2)/2)*((k+1)/(n+1))*binomial(n+1, (n-k)/2)
%o A155761 flatten([[A155761(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 06 2021
%Y A155761 Cf. A064062, A126087 (row sums).
%K A155761 easy,nonn,tabl
%O A155761 0,4
%A A155761 _Paul Barry_, Jan 26 2009