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A099028
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Euler-Seidel matrix T(k,n) with start sequence e.g.f. 2x/(1+e^(2x)), read by antidiagonals.
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4
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0, 1, 1, 0, -1, -2, -3, -3, -2, 0, 0, 3, 6, 8, 8, 25, 25, 22, 16, 8, 0, 0, -25, -50, -72, -88, -96, -96, -427, -427, -402, -352, -280, -192, -96, 0, 0, 427, 854, 1256, 1608, 1888, 2080, 2176, 2176, 12465, 12465, 12038, 11184, 9928, 8320, 6432, 4352, 2176, 0
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OFFSET
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0,6
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COMMENTS
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In an Euler-Seidel matrix, the rows are consecutive pairwise sums and the columns consecutive differences.
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LINKS
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FORMULA
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Recurrence: T(k, n) = T(k-1, n) + T(k-1, n+1).
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EXAMPLE
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Seidel matrix:
[ 0 1 -2 0 8 0 -96 0 2176 0]
[ 1 -1 -2 8 8 -96 -96 2176 2176 .]
[ 0 -3 6 16 -88 -192 2080 4352 . .]
[ -3 3 22 -72 -280 1888 6432 . . .]
[ 0 25 -50 -352 1608 8320 . . . .]
[ 25 -25 -402 1256 9928 . . . . .]
[ 0 -427 854 11184 . . . . . .]
[ -427 427 12038 . . . . . . .]
[ 0 12465 . . . . . . . .]
[12465 . . . . . . . . .]
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MATHEMATICA
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T[k_, n_] := T[k, n] = If[k == 0, SeriesCoefficient[2x/(1 + E^(2x)), {x, 0, n}] n!, T[k-1, n] + T[k-1, n+1]];
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PROG
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(Sage)
def SeidelMatrixA099028(dim):
E = matrix(ZZ, dim)
t = taylor(2*x/(1+exp(2*x)), x, 0, dim + 1)
for k in (0..dim-1):
E[0, k] = factorial(k) * t.coefficient(x, k)
R = [0]
for n in (1..dim-1):
for k in (0..dim-n-1):
E[n, k] = E[n-1, k] + E[n-1, k+1]
R.extend([E[n-k, k] for k in (0..n)])
return R
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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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