A397329 Triangle read by rows: T(n, k) = 2^(n - k) * P(n, k, 1/2) where P(n, k, x) = (1/k!) * Sum_{j=0..k}((-1)^(k - j) * binomial(k, j) * (n + x + j)^n).

1, 3, 1, 25, 12, 1, 343, 193, 27, 1, 6561, 4040, 730, 48, 1, 161051, 105121, 22230, 1970, 75, 1, 4826809, 3281908, 772891, 82880, 4355, 108, 1, 170859375, 119739649, 30506721, 3742291, 242725, 8435, 147, 1, 6975757441, 5003902800, 1353936340, 183694560, 13860966, 602448, 14868, 192, 1

Offset

0,2

Comments

For comments and a table of the polynomials see A397333, which is the main entry for this family of triangles.

Links

Table of n, a(n) for n=0..44.

Formula

T(n, k) = 2^(n - k) * t(n, k) where t(n, k) = t(n - 1, k - 1) + (n + 2*k + 1/2) * t(n - 1, k) + (n + k + 1/2) * (k + 1) * t(n - 1, k + 1) for 0 <= k < n, and t(n, n) = 1.

T(n, k) = 2^(n - k) * [x^k] b(n, n) where b(n, k) = (k + 1/2)*b(n-1, k) + x*b(n-1, k+1) if n > 0 and b(0, k) = 1.

Example

Triangle starts:
  [0]         1;
  [1]         3,         1;
  [2]        25,        12,        1;
  [3]       343,       193,       27,       1;
  [4]      6561,      4040,      730,      48,      1;
  [5]    161051,    105121,    22230,    1970,     75,    1;
  [6]   4826809,   3281908,   772891,   82880,   4355,  108,   1;
  [7] 170859375, 119739649, 30506721, 3742291, 242725, 8435, 147, 1;

Maple

t := proc(n, k) option remember; if n = k then return 1 fi; if k < 0 or k > n then return 0 fi; t(n-1, k-1) + (n + 2*k + 1/2) * t(n - 1, k) + (n + k + 1/2) * (k + 1) * t(n-1, k+1) end: T := (n, k) -> 2^(n - k) * t(n, k):
# Alternative:
P := (n, k, x) -> local j; add((-1)^(k - j) * binomial(k, j) * (n + x + j)^n , j = 0..k) / k!: T := (n, k) -> 2^(n-k) * P(n, k, 1/2): seq(seq(T(n, k), k = 0..n), n = 0..9);

Prog

(SageMath)
R.<z> = ZZ[]
@cached_function
def b(a, n, k):
    if n == 0: return R(1)
    return (k + a)*b(a, n-1, k) + z*b(a, n-1, k+1)
def T_row(a, n) -> list[int]: return [2^(n - k) * c for k, c in enumerate(b(a, n, n))]
for n in range(8): print(T_row(1/2, n))

Crossrefs

Cf. A085527 (column 1), A033428 (subdiagonal), A397328 (row sums), A397333 (p(x=1)), A397331 (p(x=-1)), A397048 (p(x=0)).

Sequence in context: A259208 A332411 A104033 * A217472 A156654 A087071

Adjacent sequences: A397326 A397327 A397328 * A397330 A397331 A397332

Keyword

nonn,tabl,new

Author

Peter Luschny, Jun 21 2026

Status

approved