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A247493 Triangle read by rows: T(n, k) = C(n, k)*C(2*k, k)/(k+1) - sum(j = 0..k, (-1)^j*(1-j)^n*C(k, j)/k!), 0<=k<=n.

%I A247493 #14 Mar 04 2017 13:52:07
%S A247493 0,0,0,0,1,1,0,2,6,4,0,3,11,22,13,0,4,20,45,75,41,0,5,29,110,190,261,
%T A247493 131,0,6,42,154,560,826,938,428,0,7,55,322,749,2646,3570,3452,1429,0,
%U A247493 8,72,335,2499,3885,12012,15198,12897,4861,0,9,89,770,650,16947,21693,53880,63915,48655,16795,0,10,110,484,11660,-8338,97482
%N A247493 Triangle read by rows: T(n, k) = C(n, k)*C(2*k, k)/(k+1) - sum(j = 0..k, (-1)^j*(1-j)^n*C(k, j)/k!), 0<=k<=n.
%C A247493 First negative value appears at T(11,5). - _Indranil Ghosh_, Mar 04 2017
%H A247493 Indranil Ghosh, <a href="/A247493/b247493.txt">Rows 0..100, flattened</a>
%F A247493 A105794(n, k) = (-1)^(n-k)*(C(n, k)*Catalan(k) - T(n, k)).
%F A247493 A247491(n) = Sum(k=0..n, (-1)^(n-k+1)*T(n, k)).
%F A247493 A001453(n) = T(n, n).
%F A247493 T(n,k) = A098474 (n,k) - A105794 (n,k). - _Michel Marcus_, Mar 04 2017
%e A247493 0;
%e A247493 0, 0;
%e A247493 0, 1, 1;
%e A247493 0, 2, 6, 4;
%e A247493 0, 3, 11, 22, 13;
%e A247493 0, 4, 20, 45, 75, 41;
%e A247493 0, 5, 29, 110, 190, 261, 131;
%e A247493 0, 6, 42, 154, 560, 826, 938, 428;
%p A247493 T := proc(n, k) binomial(n,k)*binomial(2*k,k)/(k+1) - add((-1)^j*(1-j)^n /(j!*(k-j)!), j = 0..k) end:
%p A247493 for n from 0 to 12 do seq(T(n,k), k=0..n) od;
%t A247493 Flatten[Table[(Binomial[n,k] * Binomial[2k,k] / (k+1)) - Sum[(-1)^j*(1-j)^n*Binomial[k,j]/k!,{j,0,k}],{n,0,10},{k,0,n}]] (* _Indranil Ghosh_, Mar 04 2017 *)
%o A247493 (PARI)
%o A247493 tabl(nn) = {for (n=0, nn, for (k=0, n, print1((binomial(n,k)*binomial(2*k,k)/(k+1))-sum(j=0, k, (-1)^j*(1-j)^n*binomial(k,j)/k!),", ",);); print(););};
%o A247493 tabl(10); \\ _Indranil Ghosh_, Mar 04 2017
%Y A247493 Cf. A001453, A098474, A105794, A247491.
%K A247493 sign,tabl
%O A247493 0,8
%A A247493 _Peter Luschny_, Oct 02 2014