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A263985
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Triangle of signed Eulerian numbers on involutions, read by rows.
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0
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1, -1, 1, -1, -2, 1, 1, -2, -2, 1, 1, 6, 0, -2, 1, -1, 3, 14, 2, -3, 1, -1, -12, -15, 12, -1, -4, 1, 1, -4, -51, -76, 4, -3, -4, 1, 1, 20, 67, -10, -80, 30, 3, -4, 1, -1, 5, 137, 517, 414, 66, 75, 7, -5, 1, -1, -30, -192, -140, 721, 588, -49, 44, 0, -6, 1
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OFFSET
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1,5
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LINKS
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FORMULA
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T(n, k) = Sum_{m=0..k+1} (-1)^(k-m+1)*C(n+1,k-m+1)*Sum_{j=0..floor(n/2)} (-1)^j*C(C(m+1,2)+j-1,j)*C(m,n-2*j);
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EXAMPLE
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Triangle begins:
1;
-1, 1;
-1, -2, 1;
1, -2, -2, 1;
1, 6, 0, -2, 1;
-1, 3, 14, 2, -3, 1;
-1, -12, -15, 12, -1, -4, 1;
...
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MATHEMATICA
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T[n_, k_] := Sum[(-1)^(k-m+1) Binomial[n+1, k-m+1] Sum[(-1)^j Binomial[ Binomial[m+1, 2]+j-1, j] Binomial[m, n-2j], {j, 0, n/2}], {m, 0, k+1}];
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PROG
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(PARI) T(n, k) = sum(m=0, k+1, (-1)^(k-m+1)*binomial(n+1, k-m+1)*sum(j=0, n\2, (-1)^j*binomial(binomial(m+1, 2)+j-1, j)*binomial(m, n-2*j)));
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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