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A341102
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T(n, k) = [n, k] - {n, k}, where [n, k] are the (unsigned) Stirling cycle numbers and {n, k} the Stirling set numbers. Table T(n, k) read by rows, for n >= 3 and 1 <= k <= n-2.
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0
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1, 5, 4, 23, 35, 10, 119, 243, 135, 20, 719, 1701, 1323, 385, 35, 5039, 12941, 12166, 5068, 910, 56, 40319, 109329, 115099, 59514, 15498, 1890, 84, 362879, 1026065, 1163370, 689575, 226800, 40446, 3570, 120, 3628799, 10627617, 12725075, 8263750, 3170200, 722568, 93786, 6270, 165
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OFFSET
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3,2
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LINKS
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FORMULA
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T(n, k) = Sum_{j=0..k} (binomial(n+j-1, 2*k) - binomial(n+k-j, 2*k))*A340556(k, j).
E.g.f.: (1 - z)^(-x) - exp(x*(exp(z) - 1)) (unrestricted rows and n >= 0).
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EXAMPLE
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Triangle starts:
[ 3] [1]
[ 4] [5, 4]
[ 5] [23, 35, 10]
[ 6] [119, 243, 135, 20]
[ 7] [719, 1701, 1323, 385, 35]
[ 8] [5039, 12941, 12166, 5068, 910, 56]
[ 9] [40319, 109329, 115099, 59514, 15498, 1890, 84]
[10] [362879, 1026065, 1163370, 689575, 226800, 40446, 3570, 120]
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MAPLE
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# Giving full rows for n >= 0:
gf := (1 - z)^(-x) - exp(x*(exp(z) - 1));
ser := series(gf, z, 20): coeffz := n -> coeff(ser, z, n):
A341102row := n -> seq(n!*coeff(coeffz(n), x, k), k=0..n):
for n from 0 to 9 do A341102row(n) od;
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PROG
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(SageMath)
for n in (3..11):
print([stirling_number1(n, k) - stirling_number2(n, k) for k in (1..n-2)])
(PARI) T(n, k) = abs(stirling(n, k, 1)) - stirling(n, k, 2); \\ Michel Marcus, Feb 24 2021
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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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