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A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0. 16
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 (list; table; graph; refs; listen; history; text; internal format)
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

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 * A246118 A171882 A214075

Adjacent sequences:  A003989 A003990 A003991 * A003993 A003994 A003995

KEYWORD

easy,nice,nonn,tabl,changed

AUTHOR

Marc LeBrun

EXTENSIONS

More terms from David W. Wilson

Edited by Paul D. Hanna, Oct 22 2004

STATUS

approved

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Last modified November 16 22:43 EST 2018. Contains 317275 sequences. (Running on oeis4.)