A394824 - OEIS

%I #11 Apr 08 2026 10:35:26

%S 1,1,1,4,3,1,27,17,5,1,256,142,38,7,1,3125,1569,389,67,9,1,46656,

%T 21576,5016,816,104,11,1,823543,355081,78077,12085,1471,149,13,1,

%U 16777216,6805296,1424560,210968,24648,2402,202,15,1,387420489,148869153,29818917,4243743,477789,44961,3657,263,17,1

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

%F From _Peter Luschny_, Apr 08 2026: (Start)

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

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

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

%e Triangle starts:

%e   [0]         1;

%e   [1]         1,       1;

%e   [2]         4,       3,       1;

%e   [3]        27,      17,       5,      1;

%e   [4]       256,     142,      38,      7,     1;

%e   [5]      3125,    1569,     389,     67,     9,    1;

%e   [6]     46656,   21576,    5016,    816,   104,   11,   1;

%e   [7]    823543,  355081,   78077,  12085,  1471,  149,  13,  1;

%e   [8]  16777216, 6805296, 1424560, 210968, 24648, 2402, 202, 15, 1;

%p T := (n, k) -> ifelse(n = 0, 1, hypergeom([k, k - n], [], -1/n) * n^(n - k)):

%p for n from 0 to 9 do seq(simplify(T(n, k)), k = 0..n) od;

%p # Alternative:

%p T := proc(n, k) option remember;

%p     if k = n then return 1 fi;

%p     if k > n or n < 0 or k < 0 then return 0 fi;

%p     (2*k + 1)*T(n, k+1) + (k+1)*(n - k - 1)*T(n, k+2) end:

%p for n from 0 to 8 do seq(T(n, k), k = 0..n) od;

%p # Alternative:

%p egf := exp(x * (1 + y) * exp(y / (1 + y))) / (1 + y):

%p T := proc(n, k) local row, M; M := n - k;

%p     row := n! * coeff(series(egf, x, n + 1), x, n);

%p     return M! * coeff(series(row, y, M + 1), y, M);

%p end:

%p for n from 0 to 8 do seq(T(n, k), k = 0..n) od;

%t T[0, 0] = 1; T[n_, m_] = n^(n - m) HypergeometricPFQ[{m - n, m}, {}, -n^(-1)]

%t Table[T[n, m], {n, 0, 8}, {m, 0, n}] // MatrixForm

%o (Python)

%o def A394824row(n) -> list[int]:

%o     row = [0] * (n + 1); row[n] = 1

%o     for k in range(n, 0, -1):

%o         row[k - 1] = (2 * k - 1) * row[k] + k * (n - k) * (row[k + 1] if k + 1 <= n else 0)

%o     return row

%o for n in range(0, 9): print(A394824row(n))  # _Peter Luschny_, Apr 08 2026

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

%K nonn,tabl

%O 0,4

%A _Peter Luschny_, Apr 03 2026