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A236935
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The infinite Seidel matrix H read by antidiagonals upwards; H = (H(n,k): n,k >= 0).
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2
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1, 0, -1, -1, -1, 0, 0, 1, 2, 2, 5, 5, 4, 2, 0, 0, -5, -10, -14, -16, -16, -61, -61, -56, -46, -32, -16, 0, 0, 61, 122, 178, 224, 256, 272, 272, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0, 0, -1385, -2770, -4094, -5296, -6320, -7120, -7664, -7936, -7936, -50521, -50521, -49136, -46366, -42272, -36976, -30656, -23536, -15872, -7936, 0
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OFFSET
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0,9
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COMMENTS
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This is, in essence, a signed version of the triangle in A008280.
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LINKS
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FORMULA
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H(n,k) = Sum_{i=0..n} binomial(n,i)*H(0,k+i).
H(n,k) = Sum_{i=0..k} (-1)^i*binomial(k,i)*H(n+k-i,0).
Bivariate e.g.f.: Sum_{n,k>=0} H(n,k)*(x^n/n!)*(y^k/k!) = 2*exp(x)/(1 + exp(2*(x+y))).
H(n,k) = (-1)^(n+k)*A239005(n+k,k), where the latter is a triangle.
H(n,k) = -A008280(n+k,k) if ((n+k) mod 4) == 1 or 2, and H(n,k) = A008280(n+k,k) if ((n+k) mod 4) == 3 or 0, provided A008280 is read as a triangle. (End)
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EXAMPLE
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Array begins:
1 -1 0 2 0 -16 0 272 0 ...
0 -1 2 2 -16 -16 272 272 ...
-1 1 4 -14 -32 256 544 ...
0 5 -10 -46 224 800 ...
5 -5 -56 178 1024 ...
0 -61 122 1202 ...
-61 61 1324 ...
0 1385 ...
1385 ...
...
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PROG
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(PARI) a(n) = 2^n*2^(n+1)*(subst(bernpol(n+1, x), x, 3/4) - subst(bernpol(n+1, x), x, 1/4))/(n+1) /* A122045 */
H(n, k) = sum(i=0, k, (-1)^i*binomial(k, i)*a(n+k-i)) /* Petros Hadjicostas, Feb 21 2021 */
/* Second PARI program (same a(n) for A122045 as above) */
H(n, k) = (-1)^(n+k)*sum(i=0, k, binomial(k, i)*a(n+i)) /* Petros Hadjicostas, Feb 21 2021 */
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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