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 A119328 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)}}. 4
 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, 1, -10, 33, -44, 25, -6, 1, 0, -1, 12, -51, 96, -85, 36, -7, 1, 0, 1, -14, 73, -180, 225, -146, 49, -8, 1, 0, -1, 16, -99, 304, -501, 456, -231, 64, -9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS Row sums are A021913(n+2). Product with Pascal's triangle A007318 is A119326. A signed version of A058716. LINKS FORMULA Column k has g.f. (x/(1+x))^k*sum{j=0..k, C(k,2j)x^(2j)} EXAMPLE Triangle begins 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, 1, -10, 33, -44, 25, -6, 1, 0, -1, 12, -51, 96, -85, 36, -7, 1, 0, 1, -14, 73, -180, 225, -146, 49, -8, 1, 0, -1, 16, -99, 304, -501, 456, -231, 64, -9, 1 MATHEMATICA 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 *) CROSSREFS Sequence in context: A301422 A055340 A058716 * A048723 A088455 A004248 Adjacent sequences:  A119325 A119326 A119327 * A119329 A119330 A119331 KEYWORD easy,sign,tabl AUTHOR Paul Barry, May 14 2006 STATUS approved

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Last modified August 14 19:45 EDT 2020. Contains 336483 sequences. (Running on oeis4.)