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 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Paolo Xausa, Table of n, a(n) for n = 0..11475 (antidiagonals 0..150 of the square, flattened). 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. EXAMPLE 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] 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 *) PROG (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)]) CROSSREFS The matrix transpose of A326323. Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324, A000027, A028872. Sequence in context: A139329 A335432 A229557 * A256268 A213275 A069777 Adjacent sequences: A332697 A332698 A332699 * A332701 A332702 A332703 KEYWORD nonn,tabl AUTHOR Peter Luschny, Feb 28 2020 STATUS approved

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Last modified August 14 12:36 EDT 2024. Contains 375164 sequences. (Running on oeis4.)