A394813 Triangle read by rows: T(n, k) is the k-th even-indexed column of the array A394582.

1, 2, 1, 3, 10, 1, 4, 42, 42, 1, 5, 120, 441, 170, 1, 6, 275, 2508, 4224, 682, 1, 7, 546, 10010, 45760, 39039, 2730, 1, 8, 980, 31668, 307450, 784212, 355446, 10922, 1, 9, 1632, 84966, 1508988, 8666515, 13021320, 3215397, 43690, 1

Offset

1,2

Comments

This triangle represents the scaling phase of the parity-dependent recurrence defined in A394582. It is the counterpart to the central factorial triangle A008957.

Links

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).

Formula

T(n, k) = A394582(n - k + 1, 2*k) for 1 <= k <= n.

T(n, 1) = n, T(n, n) = 1;

T(n, k) = (n-k+1)^2 * T(n-1, k-1) + (n-k+1) * Sum_{j=1..n-k} (j * T(j+k-2, k-1)) for 1 < k < n.

T(n, k) = ((n - k + 1) / (2*n - 2*k + 1)!) * Sum_{i=0..2*n - 2*k + 1} (-1)^(2*n - 2*k + 1 - i) * binomial(2*n - 2*k + 1, i) * (i - n + k)^(2*n - 1).

T(n, k) = (n - k + 1) * Sum_{j=0..2*n - 1} binomial(2*n - 1, j) * (k - n)^(2*n - 1 - j) * Stirling2(j, 2*n - 2*k + 1).

From Peter Luschny, Apr 02 2026: (Start)

T(n, k) = 2*m*Sum_{j=0..n-k} (-1)^j*(m - j)^(2*n)/((2*m - j)!*j!) where m = n - k + 1.

T(n, k) = Sum_{j=0..n-k} ((-1)^j*binomial(2*m, j)*(m-j)^(2*n))/(2*m-1)! where m = n - k + 1.

T(n, k) = ((m * T(n - 1, k)) / (n - k)) + (m^2 * T(n - 1, k - 1)) where m = n - k + 1, with boundary conditions T(n, 1) = n and T(n, n) = 1. (End)

T(n, k) = A395021(2*n, 2*n-2*k+1).

T(n, k) = ((2*n)! / (2*n-2*k+1)!) * [x^(2*n)] ( (2*sinh(x/2))^(2*n-2*k+1) * exp(x/2) ).

Example

The triangle starts:
  1;
  2,   1;
  3,  10,     1;
  4,  42,    42,      1;
  5, 120,   441,    170,      1;
  6, 275,  2508,   4224,    682,      1;
  7, 546, 10010,  45760,  39039,   2730,     1;
  8, 980, 31668, 307450, 784212, 355446, 10922, 1;

Maple

T := proc(n, k) local m; m := n-k+1; 2*m*sum((-1)^j*(m-j)^(2*n)/((2*m-j)!*j!), j=0..n-k) end:
for n from 1 to 9 do seq(T(n, k), k = 1..n) od;

Mathematica

A394813[n_, k_] := A394813[n, k] = Switch[k, 1, n, n, 1, _, #*A394813[n-1, k]/(n-k) + #^2*A394813[n-1, k-1] & [n-k+1]];
Table[A394813[n, k], {n, 10}, {k, n}] (* Paolo Xausa, Apr 04 2026 *)

Prog

(Python)
from functools import cache
@cache
def T(n: int, k: int) -> int:
    if k == 1: return n
    if n == k: return 1
    return (((n - k + 1) * T(n - 1, k)) // (n - k)) + (((n - k + 1) ** 2) * T(n - 1, k - 1))
for n in range(1, 10): print([T(n, k) for k in range(1, n + 1)])  # Peter Luschny, Apr 02 2026

Crossrefs

Cf. A394582 (Parent array, of which this triangle is the even-column component).

Cf. A008957 (The odd-column counterpart; together they form A394582).

Diagonals: A000012 (1st diagonal), A020988 (2nd diagonal), A002451 (3 * diagonal 3), A383838 (4 * diagonal 4), A383840 (5 * diagonal 5).

Columns: A000027 (column 1), A108678 (column 2).

Cf. A036969, A395021.

Sequence in context: A301282 A246063 A332064 * A229417 A337890 A337888

Adjacent sequences: A394810 A394811 A394812 * A394814 A394815 A394816

Keyword

nonn,tabl

Author

Husiev Andrii Alekseevich, Apr 02 2026

Status

approved