A394824 Triangle read by rows: T(n, k) = n^(n - k) * hypergeom([k, k-n], [], -1/n) for n > 0, T(0, 0) = 1.

1, 1, 1, 4, 3, 1, 27, 17, 5, 1, 256, 142, 38, 7, 1, 3125, 1569, 389, 67, 9, 1, 46656, 21576, 5016, 816, 104, 11, 1, 823543, 355081, 78077, 12085, 1471, 149, 13, 1, 16777216, 6805296, 1424560, 210968, 24648, 2402, 202, 15, 1, 387420489, 148869153, 29818917, 4243743, 477789, 44961, 3657, 263, 17, 1

Offset

0,4

Links

Table of n, a(n) for n=0..54.

Formula

From Peter Luschny, Apr 08 2026: (Start)

T(n, k) = (2*k + 1)*T(n, k+1) + (k+1)*(n - k - 1)*T(n, k+2).

Let egf(x, y) = exp(x * (1 + y) * exp(y / (1 + y))) / (1 + y), then

T(n, k) = n! * (n-k)! * [x^n * y^(n-k)] egf(x, y). (End)

Example

Triangle starts:
  [0]         1;
  [1]         1,       1;
  [2]         4,       3,       1;
  [3]        27,      17,       5,      1;
  [4]       256,     142,      38,      7,     1;
  [5]      3125,    1569,     389,     67,     9,    1;
  [6]     46656,   21576,    5016,    816,   104,   11,   1;
  [7]    823543,  355081,   78077,  12085,  1471,  149,  13,  1;
  [8]  16777216, 6805296, 1424560, 210968, 24648, 2402, 202, 15, 1;

Maple

T := (n, k) -> ifelse(n = 0, 1, hypergeom([k, k - n], [], -1/n) * n^(n - k)):
for n from 0 to 9 do seq(simplify(T(n, k)), k = 0..n) od;
# Alternative:
T := proc(n, k) option remember;
    if k = n then return 1 fi;
    if k > n or n < 0 or k < 0 then return 0 fi;
    (2*k + 1)*T(n, k+1) + (k+1)*(n - k - 1)*T(n, k+2) end:
for n from 0 to 8 do seq(T(n, k), k = 0..n) od;
# Alternative:
egf := exp(x * (1 + y) * exp(y / (1 + y))) / (1 + y):
T := proc(n, k) local row, M; M := n - k;
    row := n! * coeff(series(egf, x, n + 1), x, n);
    return M! * coeff(series(row, y, M + 1), y, M);
end:
for n from 0 to 8 do seq(T(n, k), k = 0..n) od;

Mathematica

T[0, 0] = 1; T[n_, m_] = n^(n - m) HypergeometricPFQ[{m - n, m}, {}, -n^(-1)]
Table[T[n, m], {n, 0, 8}, {m, 0, n}] // MatrixForm

Prog

(Python)
def A394824row(n) -> list[int]:
    row = [0] * (n + 1); row[n] = 1
    for k in range(n, 0, -1):
        row[k - 1] = (2 * k - 1) * row[k] + k * (n - k) * (row[k + 1] if k + 1 <= n else 0)
    return row
for n in range(0, 9): print(A394824row(n))  # Peter Luschny, Apr 08 2026

Crossrefs

Cf. A000312 (column 0), A001865 (column 1).

Sequence in context: A208057 A298673 A245732 * A039621 A142158 A203412

Adjacent sequences: A394821 A394822 A394823 * A394825 A394826 A394827

Keyword

nonn,tabl

Author

Peter Luschny, Apr 03 2026

Status

approved