A332700 A(n, k) = Sum_{j=0..n} j!*Stirling2(n, j)*(k-1)^(n-j), for n >= 0 and k >= 0, read by ascending antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 13, 4, 1, 1, 1, 120, 75, 22, 5, 1, 1, 1, 720, 541, 160, 33, 6, 1, 1, 1, 5040, 4683, 1456, 285, 46, 7, 1, 1, 1, 40320, 47293, 15904, 3081, 456, 61, 8, 1, 1, 1, 362880, 545835, 202672, 40005, 5656, 679, 78, 9, 1, 1

Offset: 0

Author

Peter Luschny, Feb 28 2020

Keywords

• nonn
• tabl

Examples

			Array begins:
[0] 1, 1,       1,       1,        1,         1,         1, ...    A000012
[1] 1, 1,       1,       1,        1,         1,         1, ...    A000012
[2] 1, 2,       3,       4,        5,         6,         7, ...    A000027
[3] 1, 6,       13,      22,       33,        46,        61, ...   A028872
[4] 1, 24,      75,      160,      285,       456,       679, ...
[5] 1, 120,     541,     1456,     3081,      5656,      9445, ...
[6] 1, 720,     4683,    15904,    40005,     84336,     158095, ...
[7] 1, 5040,    47293,   202672,   606033,    1467376,   3088765, ...
[8] 1, 40320,   545835,  2951680,  10491885,  29175936,  68958295, ...
[9] 1, 362880,  7087261, 48361216, 204343641, 652606336, 1731875605, ...
       A000142, A000670, A122704,  A255927,   A326324, ...
Seen as a triangle:
[0] [1]
[1] [1, 1]
[2] [1, 1,     1]
[3] [1, 2,     1,     1]
[4] [1, 6,     3,     1,     1]
[5] [1, 24,    13,    4,     1,    1]
[6] [1, 120,   75,    22,    5,    1,   1]
[7] [1, 720,   541,   160,   33,   6,   1,  1]
[8] [1, 5040,  4683,  1456,  285,  46,  7,  1, 1]
[9] [1, 40320, 47293, 15904, 3081, 456, 61, 8, 1, 1]
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (antidiagonals 0..150 of the square, flattened).

Crossrefs

The matrix transpose of A326323.

Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324, A000027, A028872.

Programs

Maple

# Prints array by row.
A := (n, k) -> add(combinat:-eulerian1(n, j)*k^j, j=0..n):
seq(print(seq(A(n,k), k=0..10)), n=0..8);
# Alternative:
egf := n -> `if`(n=1, 1/(1-x), (n-1)/(n - exp((n-1)*x))):
ser := n -> series(egf(n), x, 21):
for n from 0 to 6 do seq(n!*coeff(ser(k), x, n), k=0..9) od;
# Or:
A := (n, k) -> if k = 0 or n = 0 then 1 elif k = 1 then n! else
polylog(-n, 1/k)*(k-1)^(n+1)/k fi:
for n from 0 to 6 do seq(A(n, k), k=0..9) od;

Mathematica

A332700[n_, k_] := n! + Sum[j! StirlingS2[n, j] (k-1)^(n-j), {j, n-1}];
Table[A332700[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 01 2024 *)

Sage

def T(n, k):
    return sum(factorial(j)*stirling_number2(n, j)*(k-1)^(n-j) for j in range(n+1))
for n in range(8): print([T(n, k) for k in range(8)])

Formula

A(n, k) = Sum_{j=0..n} E(n, j)*k^j, where E(n, k) = A173018(n, k).

A(n, 1) = n!*[x^n] 1/(1-x).

A(n, k) = n!*[x^n] (k-1)/(k - exp((k-1)*x)) for k != 1.

A(n, k) = PolyLog(-n, 1/k)*(k-1)^(n+1)/k for k >= 2.