A394444 Array read by ascending antidiagonals: A(n, k) = Sum_{j=0..k} (-1)^(k - j) * binomial(k, j) * j^n * 2^j.

1, 0, 1, 0, 2, 1, 0, 2, 4, 1, 0, 2, 12, 6, 1, 0, 2, 28, 30, 8, 1, 0, 2, 60, 126, 56, 10, 1, 0, 2, 124, 462, 344, 90, 12, 1, 0, 2, 252, 1566, 1880, 730, 132, 14, 1, 0, 2, 508, 5070, 9368, 5370, 1332, 182, 16, 1, 0, 2, 1020, 15966, 43736, 36250, 12372, 2198, 240, 18, 1

Offset

0,5

Links

Alois P. Heinz, Antidiagonals n = 0..150, flattened

Dalton Heilig, Pascal's triangle in difference tables and an alternate approach to exponential functions, Rose-Hulman Undergrad. Math. J., 18, 2017, 2, Article 4.

Formula

A(n, k) = Sum_{j=0..k} 2^j*binomial(k, j)*j!*Stirling2(n, j). (See Mikhail Kurkov's formula in A375540 and the links given there.)

More general, set A(n, k, m) = Sum_{j=0..k} (-1)^(k - j) * binomial(k, j) * j^n * m^j, then for m = 1 the Fubini array/triangle A131689 is obtained.

E.g.f. for column k: (2*e^x - 1)^k. - Dalton Heilig, Apr 22 2026

Example

Array starts:
  [0]  1, 1,   1,     1,      1,       1,       1,        1,        1, ... [A000012]
  [1]  0, 2,   4,     6,      8,      10,      12,       14,       16, ... [A005843]
  [2]  0, 2,  12,    30,     56,      90,     132,      182,      240, ... [A002939]
  [3]  0, 2,  28,   126,    344,     730,    1332,     2198,     3376, ...
  [4]  0, 2,  60,   462,   1880,    5370,   12372,    24710,    44592, ...
  [5]  0, 2, 124,  1566,   9368,   36250,  106452,   259574,   554416, ...
  [6]  0, 2, 252,  5070,  43736,  228090,  856212,  2562182,  6511920, ...
  [7]  0, 2, 508, 15966, 195224, 1359130, 6505812, 23928758, 72592816, ...
  [A000007,A040000,A173033,A366058,...]
.
Seen as a triangle, T(n, k) = A(n - k, k):
  [0]  1;
  [1]  0, 1;
  [2]  0, 2,   1;
  [3]  0, 2,   4,   1;
  [4]  0, 2,  12,   6,   1;
  [5]  0, 2,  28,  30,   8,  1;
  [6]  0, 2,  60, 126,  56, 10,  1;
  [7]  0, 2, 124, 462, 344, 90, 12, 1;

Maple

A := (n, k) -> local j; add((-1)^(k-j)*binomial(k, j)*j^n*2^j, j = 0..k):
seq(lprint([n], seq(A(n, k), k = 0..8)), n = 0..7);
# Alternative:
A:= proc(n, k) option remember;
     `if`(n=0, 1, k*(A(n-1, k)+A(n-1, k-1)))
    end:
seq(seq(A(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, Apr 28 2026

Crossrefs

Cf. A375540 (main diagonal), A394441 (row sums), A131689 (Fubini).

Rows: A000012, A005843, A002939.

Columns: A000007, A040000, A173033, A366058.

Sequence in context: A122542 A395140 A227341 * A098542 A320019 A141343

Adjacent sequences: A394441 A394442 A394443 * A394445 A394446 A394447

Keyword

nonn,tabl

Author

Peter Luschny, Mar 20 2026

Status

approved