A357119
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} |Stirling1(n,k*j)|.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 6, 0, 1, 0, 0, 3, 24, 0, 1, 0, 0, 1, 12, 120, 0, 1, 0, 0, 0, 6, 60, 720, 0, 1, 0, 0, 0, 1, 35, 360, 5040, 0, 1, 0, 0, 0, 0, 10, 226, 2520, 40320, 0, 1, 0, 0, 0, 0, 1, 85, 1645, 20160, 362880, 0, 1, 0, 0, 0, 0, 0, 15, 735, 13454, 181440, 3628800, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 0, 0, 0, 0, 0, ...
0, 2, 1, 0, 0, 0, 0, ...
0, 6, 3, 1, 0, 0, 0, ...
0, 24, 12, 6, 1, 0, 0, ...
0, 120, 60, 35, 10, 1, 0, ...
0, 720, 360, 226, 85, 15, 1, ...
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T(n, k) = sum(j=0, n, abs(stirling(n, k*j, 1)));
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T(n, k) = if(k==0, 0^n, n!*polcoef(sum(j=0, n\k, (-log(1-x+x*O(x^n)))^(k*j)/(k*j)!), n));
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Pochhammer(x, n) = prod(k=0, n-1, x+k);
T(n, k) = if(k==0, 0^n, my(w=exp(2*Pi*I/k)); round(sum(j=0, k-1, Pochhammer(w^j, n)))/k);
A357869
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* Stirling2(n,k*j)/j!.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 5, 0, 1, 0, 0, 6, 15, 0, 1, 0, 0, 6, 26, 52, 0, 1, 0, 0, 0, 36, 150, 203, 0, 1, 0, 0, 0, 24, 150, 962, 877, 0, 1, 0, 0, 0, 0, 240, 900, 6846, 4140, 0, 1, 0, 0, 0, 0, 120, 1560, 9366, 54266, 21147, 0, 1, 0, 0, 0, 0, 0, 1800, 8400, 101556, 471750, 115975, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 0, 0, 0, 0, ...
0, 2, 2, 0, 0, 0, ...
0, 5, 6, 6, 0, 0, ...
0, 15, 26, 36, 24, 0, ...
0, 52, 150, 150, 240, 120, ...
-
T(n, k) = sum(j=0, n, (k*j)!*stirling(n, k*j, 2)/j!);
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T(n, k) = if(k==0, 0^n, n!*polcoef(exp((exp(x+x*O(x^n))-1)^k), n));
A357868
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* Stirling2(n,k*j).
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 0, 2, 13, 0, 1, 0, 0, 6, 75, 0, 1, 0, 0, 6, 38, 541, 0, 1, 0, 0, 0, 36, 270, 4683, 0, 1, 0, 0, 0, 24, 150, 2342, 47293, 0, 1, 0, 0, 0, 0, 240, 1260, 23646, 545835, 0, 1, 0, 0, 0, 0, 120, 1560, 16926, 272918, 7087261, 0, 1, 0, 0, 0, 0, 0, 1800, 8400, 197316, 3543630, 102247563, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 0, 0, 0, 0, ...
0, 3, 2, 0, 0, 0, ...
0, 13, 6, 6, 0, 0, ...
0, 75, 38, 36, 24, 0, ...
0, 541, 270, 150, 240, 120, ...
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T(n, k) = sum(j=0, n, (k*j)!*stirling(n, k*j, 2));
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T(n, k) = if(k==0, 0^n, n!*polcoef(1/(1-(exp(x+x*O(x^n))-1)^k), n));
Showing 1-3 of 3 results.