A382792
a(n) = Sum_{k=0..n} (Stirling1(n,k) * k!)^2.
Original entry on oeis.org
1, 1, 5, 76, 2392, 126676, 10057204, 1114096320, 163918005696, 30894047577216, 7254176241285504, 2075722128162164736, 710883208780304954112, 287061726161439955116288, 134961239570613490548986112, 73079781978184515947237031936, 45150931601954398539342470578176
Offset: 0
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Table[Sum[(StirlingS1[n, k] k!)^2, {k, 0, n}], {n, 0, 16}]
Table[(n!)^2 SeriesCoefficient[1/(1 - Log[1 + x] Log[1 + y]), {x, 0, n}, {y, 0, n}], {n, 0, 16}]
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a(n) = sum(k=0, n, (k!*stirling(n, k, 1))^2); \\ Seiichi Manyama, Apr 05 2025
A382800
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / (1 - log(1-x) * log(1-y))^3.
Original entry on oeis.org
1, 0, 0, 0, 3, 0, 0, 3, 3, 0, 0, 6, 27, 6, 0, 0, 18, 78, 78, 18, 0, 0, 72, 282, 588, 282, 72, 0, 0, 360, 1272, 2988, 2988, 1272, 360, 0, 0, 2160, 6936, 16344, 24612, 16344, 6936, 2160, 0, 0, 15120, 44496, 101448, 175632, 175632, 101448, 44496, 15120, 0
Offset: 0
Square array begins:
1, 0, 0, 0, 0, 0, ...
0, 3, 3, 6, 18, 72, ...
0, 3, 27, 78, 282, 1272, ...
0, 6, 78, 588, 2988, 16344, ...
0, 18, 282, 2988, 24612, 175632, ...
0, 72, 1272, 16344, 175632, 1669128, ...
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a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*abs(stirling(n, j, 1)*stirling(k, j, 1)));
A382799
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / (1 - log(1-x) * log(1-y))^2.
Original entry on oeis.org
1, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 4, 14, 4, 0, 0, 12, 40, 40, 12, 0, 0, 48, 144, 260, 144, 48, 0, 0, 240, 648, 1284, 1284, 648, 240, 0, 0, 1440, 3528, 6936, 9588, 6936, 3528, 1440, 0, 0, 10080, 22608, 42744, 65928, 65928, 42744, 22608, 10080, 0, 0, 80640, 166896, 300240, 476808, 581952, 476808, 300240, 166896, 80640, 0
Offset: 0
Square array begins:
1, 0, 0, 0, 0, 0, ...
0, 2, 2, 4, 12, 48, ...
0, 2, 14, 40, 144, 648, ...
0, 4, 40, 260, 1284, 6936, ...
0, 12, 144, 1284, 9588, 65928, ...
0, 48, 648, 6936, 65928, 581952, ...
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a(n, k) = sum(j=0, min(n, k), j!*(j+1)!*abs(stirling(n, j, 1)*stirling(k, j, 1)));
A382823
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y)) ).
Original entry on oeis.org
1, 1, 1, 2, 2, 2, 6, 5, 5, 6, 24, 17, 17, 17, 24, 120, 74, 69, 69, 74, 120, 720, 394, 338, 337, 338, 394, 720, 5040, 2484, 1962, 1894, 1894, 1962, 2484, 5040, 40320, 18108, 13228, 12194, 12152, 12194, 13228, 18108, 40320, 362880, 149904, 101812, 89160, 87320, 87320, 89160, 101812, 149904, 362880
Offset: 0
Square array begins:
1, 1, 2, 6, 24, 120, ...
1, 2, 5, 17, 74, 394, ...
2, 5, 17, 69, 338, 1962, ...
6, 17, 69, 337, 1894, 12194, ...
24, 74, 338, 1894, 12152, 87320, ...
120, 394, 1962, 12194, 87320, 696076, ...
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a(n, k) = sum(j=0, min(n, k), j!^2*abs(stirling(n+1, j+1, 1)*stirling(k+1, j+1, 1)));
Showing 1-4 of 4 results.
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