A382825 -id:A382825 - cp's OEIS frontend

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)) ).

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

Seiichi Manyama, Apr 05 2025

			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, ...

Columns k=0..1 give A000142, A000774.
Main diagonal gives A382826.
Cf. A382824, A382825.
Cf. A099594, A379821.

a(n, k) = sum(j=0, min(n, k), j!^2*abs(stirling(n+1, j+1, 1)*stirling(k+1, j+1, 1)));

E.g.f.: 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y)) ).
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * |Stirling1(n+1,j+1)| * |Stirling1(k+1,j+1)|.

A382824 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))^2 ).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 6, 8, 8, 6, 24, 28, 34, 28, 24, 120, 124, 150, 150, 124, 120, 720, 668, 768, 854, 768, 668, 720, 5040, 4248, 4584, 5204, 5204, 4584, 4248, 5040, 40320, 31176, 31512, 35188, 37556, 35188, 31512, 31176, 40320, 362880, 259488, 246072, 265896, 290380, 290380, 265896, 246072, 259488, 362880

Offset: 0

Seiichi Manyama, Apr 05 2025

			Square array begins:
    1,   1,    2,     6,     24,     120, ...
    1,   3,    8,    28,    124,     668, ...
    2,   8,   34,   150,    768,    4584, ...
    6,  28,  150,   854,   5204,   35188, ...
   24, 124,  768,  5204,  37556,  290380, ...
  120, 668, 4584, 35188, 290380, 2546852, ...

Main diagonal gives A382827.

Cf. A382823, A382825.

a(n, k) = sum(j=0, min(n, k), j!*(j+1)!*abs(stirling(n+1, j+1, 1)*stirling(k+1, j+1, 1)));

E.g.f.: 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y))^2 ).
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} j! * (j+1)! * |Stirling1(n+1,j+1)| * |Stirling1(k+1,j+1)|.

A382828 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling1(n+1,k+1)^2.

Original entry on oeis.org

1, 4, 55, 1623, 82116, 6302028, 680105112, 98011315608, 18163969766592, 4205977241171328, 1189459906531372224, 403300593144673493184, 161454763431242385682176, 75337361633768810384542464, 40524573487904551618353921024, 24890567631479746511661428751360

Offset: 0

Seiichi Manyama, Apr 06 2025

nonn

Main diagonal of A382825.

Cf. A382676, A382806.

a(n) = sum(k=0, n, k!^2*binomial(k+2, 2)*stirling(n+1, k+1, 1)^2);

a(n) = (n!)^2 * [(x*y)^n] 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y))^3 ).

a(n) = (n!)^2 * [(x*y)^n] 1 / ( (1+x) * (1+y) * (1 - log(1+x) * log(1+y))^3 ).

cp's OEIS Frontend

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)) ).

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A382824 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))^2 ).

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PARI

Formula

A382828 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling1(n+1,k+1)^2.

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