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

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).

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

A051128 Table T(n,k) = n^k read by upwards antidiagonals (n >= 1, k >= 1).

1, 2, 1, 3, 4, 1, 4, 9, 8, 1, 5, 16, 27, 16, 1, 6, 25, 64, 81, 32, 1, 7, 36, 125, 256, 243, 64, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 11, 100, 729, 4096, 16807, 46656, 78125, 65536, 19683, 1024, 1

Offset

1,2

Comments

Sum of antidiagonals is A003101(n) for n>0. - Alford Arnold, Jan 14 2007

Links

T. D. Noe, Rows n=1..50 of triangle, flattened

G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], 2003.

Formula

a(n) = A004736(n)^A002260(n) or ((t*t+3*t+4)/2-n)^(n-(t*(t+1)/2)), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 14 2012

Example

Table begins
1,    1,    1,    1,    1, ...
2,    4,    8,   16,   32, ...
3,    9,   27,   81,  243, ...
4,   16,   64,  256, 1024, ...

Maple

A051128 := proc(n)       # Boris Putievskiy's formula
a := floor((sqrt(8*n-7)+1)/2);
b := (a+a^2)/2-n;
c := (a-a^2)/2+n;
(b+1)^c end:
seq(A051128(n), n=1..61);  # Peter Luschny, Dec 14 2012
# second Maple program:
T:= (n, k)-> n^k:
seq(seq(T(1+d-k, k), k=1..d), d=1..11);  # Alois P. Heinz, Apr 18 2020

Mathematica

Table[n^(k - n + 1), {k, 1, 11}, {n, k, 1, -1}] // Flatten (* Jean-François Alcover, Dec 14 2012 *)

Prog

(PARI) T(n, k) = n^k \\ Charles R Greathouse IV, Feb 09 2017

Crossrefs

Cf. A051129, A003992, A004248, A033918, A095891, A220415, A220416, A220417, A003101.

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Dec 11 1999

Status

approved

A009999 Triangle in which j-th entry in i-th row is (i+1-j)^j, 0<=j<=i.

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

Offset

0,5

Comments

T(n,k) is the number of ways of placing 1..k in n boxes such that each box contains at most one number, and numbers in adjacent boxes are in increasing order. This can be proved by observing that there are n-(k-1) ways of extending each of T(n-1,k-1). - Jimin Park, Apr 16 2023

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 24.

Links

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Formula

T(n,0) = 1; T(n,k) = (n-k+1)*T(n-1,k-1) for k=1..n. - Reinhard Zumkeller, Feb 02 2014

T(n,k) = Sum_{i=0..k} binomial(k,i)*T(n-1-i,k-i). - Jimin Park, Apr 16 2023

Example

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

Maple

A009999 := proc(i, j) (i+1-j)^j ; end proc: # R. J. Mathar, Jan 16 2011

Mathematica

Table[(i+1-j)^j, {i, 0, 10}, {j, 0, i}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

Prog

(Haskell)
a009999 n k = (n + 1 - k) ^ k
a009999_row n = a009999_tabl !! n
a009999_tabl = [1] : map snd (iterate f ([1, 1], [1, 1])) where
   f (us@(u:_), vs) = (us', 1 : zipWith (*) us' vs)
                      where us' = (u + 1) : us
-- Reinhard Zumkeller, Feb 02 2014

Crossrefs

Row sums give A026898.

T(2n,n) gives A000169(n+1).

Cf. A095884, A051128.

Cf. A009998 (mirrored).

Keyword

tabl,nonn,easy

Author

N. J. A. Sloane

Extensions

T(10,8) corrected by Reinhard Zumkeller, Feb 02 2014

Status

approved

A055208 Table read by ascending antidiagonals: T(n,k) (n >= 1, k >= 1) is the sum of k-th powers of digits of n.

1, 2, 1, 3, 4, 1, 4, 9, 8, 1, 5, 16, 27, 16, 1, 6, 25, 64, 81, 32, 1, 7, 36, 125, 256, 243, 64, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 1, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 2, 1, 729, 4096, 16807, 46656, 78125

Offset

1,2

Links

Table of n, a(n) for n=1..62.

Example

T(34,2) = 3^2 + 4^2 = 25.
From Georg Fischer, Mar 01 2022: (Start)
Array T(n, k) (n >= 1, k >= 1) begins:
  1,   1,   1,   1, ...
  2,   4,   8,  16, ...
  3,   9,  27,  64, ...
  4,  16,  64, 256, ...
  ...
(End)

Crossrefs

Cf. A051128. Rows include A003132, A055012, A055013, A055014. Columns include A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A000012, A007395, A000051, A034472, A052539, A034474.

Keyword

base,easy,nonn,tabl

Author

Henry Bottomley, Jun 19 2000

Extensions

Definition clarified by Georg Fischer, Mar 01 2022

Status

approved