cp's OEIS Frontend

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.

%I A332700 #20 Feb 01 2024 08:21:16
%S A332700 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,
%T A332700 541,160,33,6,1,1,1,5040,4683,1456,285,46,7,1,1,1,40320,47293,15904,
%U A332700 3081,456,61,8,1,1,1,362880,545835,202672,40005,5656,679,78,9,1,1
%N 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.
%H A332700 Paolo Xausa, <a href="/A332700/b332700.txt">Table of n, a(n) for n = 0..11475</a> (antidiagonals 0..150 of the square, flattened).
%F A332700 A(n, k) = Sum_{j=0..n} E(n, j)*k^j, where E(n, k) = A173018(n, k).
%F A332700 A(n, 1) = n!*[x^n] 1/(1-x).
%F A332700 A(n, k) = n!*[x^n] (k-1)/(k - exp((k-1)*x)) for k != 1.
%F A332700 A(n, k) = PolyLog(-n, 1/k)*(k-1)^(n+1)/k for k >= 2.
%e A332700 Array begins:
%e A332700 [0] 1, 1,       1,       1,        1,         1,         1, ...    A000012
%e A332700 [1] 1, 1,       1,       1,        1,         1,         1, ...    A000012
%e A332700 [2] 1, 2,       3,       4,        5,         6,         7, ...    A000027
%e A332700 [3] 1, 6,       13,      22,       33,        46,        61, ...   A028872
%e A332700 [4] 1, 24,      75,      160,      285,       456,       679, ...
%e A332700 [5] 1, 120,     541,     1456,     3081,      5656,      9445, ...
%e A332700 [6] 1, 720,     4683,    15904,    40005,     84336,     158095, ...
%e A332700 [7] 1, 5040,    47293,   202672,   606033,    1467376,   3088765, ...
%e A332700 [8] 1, 40320,   545835,  2951680,  10491885,  29175936,  68958295, ...
%e A332700 [9] 1, 362880,  7087261, 48361216, 204343641, 652606336, 1731875605, ...
%e A332700        A000142, A000670, A122704,  A255927,   A326324, ...
%e A332700 Seen as a triangle:
%e A332700 [0] [1]
%e A332700 [1] [1, 1]
%e A332700 [2] [1, 1,     1]
%e A332700 [3] [1, 2,     1,     1]
%e A332700 [4] [1, 6,     3,     1,     1]
%e A332700 [5] [1, 24,    13,    4,     1,    1]
%e A332700 [6] [1, 120,   75,    22,    5,    1,   1]
%e A332700 [7] [1, 720,   541,   160,   33,   6,   1,  1]
%e A332700 [8] [1, 5040,  4683,  1456,  285,  46,  7,  1, 1]
%e A332700 [9] [1, 40320, 47293, 15904, 3081, 456, 61, 8, 1, 1]
%p A332700 # Prints array by row.
%p A332700 A := (n, k) -> add(combinat:-eulerian1(n, j)*k^j, j=0..n):
%p A332700 seq(print(seq(A(n,k), k=0..10)), n=0..8);
%p A332700 # Alternative:
%p A332700 egf := n -> `if`(n=1, 1/(1-x), (n-1)/(n - exp((n-1)*x))):
%p A332700 ser := n -> series(egf(n), x, 21):
%p A332700 for n from 0 to 6 do seq(n!*coeff(ser(k), x, n), k=0..9) od;
%p A332700 # Or:
%p A332700 A := (n, k) -> if k = 0 or n = 0 then 1 elif k = 1 then n! else
%p A332700 polylog(-n, 1/k)*(k-1)^(n+1)/k fi:
%p A332700 for n from 0 to 6 do seq(A(n, k), k=0..9) od;
%t A332700 A332700[n_, k_] := n! + Sum[j! StirlingS2[n, j] (k-1)^(n-j), {j, n-1}];
%t A332700 Table[A332700[n-k, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Feb 01 2024 *)
%o A332700 (Sage)
%o A332700 def T(n, k):
%o A332700     return sum(factorial(j)*stirling_number2(n, j)*(k-1)^(n-j) for j in range(n+1))
%o A332700 for n in range(8): print([T(n, k) for k in range(8)])
%Y A332700 The matrix transpose of A326323.
%Y A332700 Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324, A000027, A028872.
%K A332700 nonn,tabl
%O A332700 0,8
%A A332700 _Peter Luschny_, Feb 28 2020