A395021 Triangle read by rows: T(n, k) = (n! / k!) * [x^n] ( (2*sinh(x/2))^k * (exp(x/2))^(k mod 2) ).

1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 5, 0, 1, 1, 1, 10, 5, 3, 1, 1, 0, 21, 0, 14, 0, 1, 1, 1, 42, 21, 42, 14, 4, 1, 1, 0, 85, 0, 147, 0, 30, 0, 1, 1, 1, 170, 85, 441, 147, 120, 30, 5, 1, 1, 0, 341, 0, 1408, 0, 627, 0, 55, 0, 1, 1, 1, 682, 341, 4224, 1408, 2508, 627, 275, 55, 6, 1

Offset

1,9

Comments

T(n, k) represents the normalized central finite differences f^k_i / k! of the power sequence f(j) = j^n with alternating indices i in {1/2, 0}.

The n-th row of this triangle is extracted from the central symmetry axis of the finite difference table of the sequence m^n. Following the standard notation f^k_i, the terms are selected using alternating indices: i = 1/2 for odd k and i = 0 for even k. All values are normalized by dividing by k! to yield the integer coefficients.

Odd-indexed columns correspond to the array A394582.

Links

Table of n, a(n) for n=1..78.

Andrii Husiev, Extended Central Factorial Numbers and the Flickering Operator, arXiv:2605.06689 [math.GM], 2026. See pp. 1-3, 7, 12-15.

Formula

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

T(n, k) = [x^n] (x^k + (k mod 2) * ((k+1)/2) * x^(k+1)) / Product_{j=1..floor((k+1)/2)} (1 - j^2 * x^2).

Boundary conditions: T(n, 1) = 1, T(n, n) = 1.

Recursive rules for 1 < k < n:

If k is even and n is odd: T(n, k) = 0.

If k is even and n is even: T(n, k) = 2 * T(n, k-1) / k.

If k is odd and n is even: T(n, k) = T(n-1, k) * (k+1) / 2.

If k is odd and n is odd: T(n, k) = T(n-1, k) * (k+1) / 2 + T(n-1, k-2) * 2 / (k-1).

Example

Triangle begins:
  n=1:  1;
  n=2:  1,  1;
  n=3:  1,  0,    1;
  n=4:  1,  1,    2,   1;
  n=5:  1,  0,    5,   0,    1;
  n=6:  1,  1,   10,   5,    3,    1;
  n=7:  1,  0,   21,   0,   14,    0,    1;
  n=8:  1,  1,   42,  21,   42,   14,    4,   1;
  n=9:  1,  0,   85,   0,  147,    0,   30,   0,  1;
  n=10: 1,  1,  170,  85,  441,  147,  120,  30,  5,  1;

Maple

egf_col := k -> (exp(x/2) - exp(-x/2))^k * exp(x/2)^(k mod 2):
ser_col := k -> series(egf_col(k), x, 12):
T := (n, k) -> coeff(ser_col(k), x, n) * n! / k!:
for n from 1 to 10 do seq(T(n, k), k = 1..n) od;  # Peter Luschny, Apr 10 2026

Prog

(Python)
from functools import cache
@cache
def T(n, k):
    if k == 1 or k == n:
        return 1
    if k < 1 or k > n:
        return 0
    if k % 2 == 0:
        if n % 2 != 0:
            return 0
        else:
            return (2 * T(n, k-1)) // k
    else:
        if n % 2 == 0:
            return T(n-1, k) * (k + 1) // 2
        else:
            term1 = T(n-1, k) * (k + 1) // 2
            term2 = T(n-1, k-2) * 2 // (k - 1)
            return term1 + term2
all_data = []
for n in range(1, 13):
    row = [T(n, k) for k in range(1, n + 1)]
    all_data.extend(row)
    print(f"[{n:>2}]", " ".join(f"{value:>4}" for value in row))
print(", ".join(map(str, all_data)))

Crossrefs

Cf. A394813 (even columns), A394582, A008957, A395022 (row sums).

Sequence in context: A077875 A198237 A122049 * A238802 A229892 A064879

Adjacent sequences: A395018 A395019 A395020 * A395022 A395023 A395024

Keyword

nonn,tabl,easy

Author

Husiev Andrii Alekseevich, Apr 10 2026

Status

approved