A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 8, 1, 0, 1, 5, 16, 27, 16, 1, 0, 1, 6, 25, 64, 81, 32, 1, 0, 1, 7, 36, 125, 256, 243, 64, 1, 0, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0

Offset

0,8

Comments

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020

T(n,k) is the number of k-letter strings over an n-element alphabet. - Stefano Spezia, Mar 15 2026

References

Miklos Bona, Introduction to Enumerative and Analytic Combinatorics, CRC Press, 2025, p. 8.

Links

T. D. Noe, Rows n = 0..50 of triangle, flattened

Formula

E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004

E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006

Example

Rows begin:
  [1, 0,  0,   0,    0,     0,      0,      0, ...],
  [1, 1,  1,   1,    1,     1,      1,      1, ...],
  [1, 2,  4,   8,   16,    32,     64,    128, ...],
  [1, 3,  9,  27,   81,   243,    729,   2187, ...],
  [1, 4, 16,  64,  256,  1024,   4096,  16384, ...],
  [1, 5, 25, 125,  625,  3125,  15625,  78125, ...],
  [1, 6, 36, 216, 1296,  7776,  46656, 279936, ...],
  [1, 7, 49, 343, 2401, 16807, 117649, 823543, ...],
  ...

Mathematica

Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten

Prog

(PARI) T(n, k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017
(Magma) [[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018

Crossrefs

Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.

Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.

Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.

Cf. A099555.

Transpose is A004248. See A051128, A095884, A009999 for other versions.

Cf. A277504 (unoriented), A293500 (chiral).

Sequence in context: A213276 A210391 A071921 * A337161 A246118 A171882

Adjacent sequences: A003989 A003990 A003991 * A003993 A003994 A003995

Keyword

easy,nice,nonn,tabl

Author

Marc LeBrun

Extensions

More terms from David W. Wilson

Edited by Paul D. Hanna, Oct 22 2004

Status

approved