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%I #10 Aug 16 2022 10:46:36
%S 1,0,1,0,-1,1,0,1,-2,1,0,-1,4,-3,1,0,1,-6,9,-4,1,0,-1,8,-19,16,-5,1,0,
%T 1,-10,33,-44,25,-6,1,0,-1,12,-51,96,-85,36,-7,1,0,1,-14,73,-180,225,
%U -146,49,-8,1,0,-1,16,-99,304,-501,456,-231,64,-9,1
%N Number triangle T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i-k, C(k,2j)*C(i-k,2j)}}.
%C Row sums are A021913(n+2). Product with Pascal's triangle A007318 is A119326.
%F Column k has g.f. (x/(1+x))^k*sum{j=0..k, C(k,2j)x^(2j)}
%e Triangle begins
%e 1,
%e 0, 1,
%e 0, -1, 1,
%e 0, 1, -2, 1,
%e 0, -1, 4, -3, 1,
%e 0, 1, -6, 9, -4, 1,
%e 0, -1, 8, -19, 16, -5, 1,
%e 0, 1, -10, 33, -44, 25, -6, 1,
%e 0, -1, 12, -51, 96, -85, 36, -7, 1,
%e 0, 1, -14, 73, -180, 225, -146, 49, -8, 1,
%e 0, -1, 16, -99, 304, -501, 456, -231, 64, -9, 1
%t t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n, i]*Sum[Binomial[k, 2 j]*Binomial[i - k, 2 j], {j, 0, i - k}], {i, 0, n}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 25 2013 *)
%K easy,sign,tabl
%O 0,9
%A _Paul Barry_, May 14 2006
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