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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

Table of n, a(n) for n=0..65.

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-Fran├žois 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 December 11 01:07 EST 2019. Contains 329910 sequences. (Running on oeis4.)