A009998 Triangle in which j-th entry in i-th row is (j+1)^(i-j).

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

Offset

0,5

Comments

Read as a square array this is the Hilbert transform of triangle A123125 (see A145905 for the definition of this term). For example, the fourth row of A123125 is (0,1,4,1) and the expansion (x + 4*x^2 + x^3)/(1-x)^4 = x + 8*x^2 + 27*x^3 + 64*x^4 + ... generates the entries in the fourth row of this array read as a square. - Peter Bala, Oct 28 2008

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

T. D. Noe, Rows n=0..50 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].

Peter Luschny, Figurate number - a very short introduction

Formula

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

T(n,m) = (m+1)*Sum_{k=0..n-m}((n+1)^(k-1)*(n-m)^(n-m-k)*(-1)^(n-m-k)*binomial(n-m-1,k-1)). - Vladimir Kruchinin, Sep 12 2015

Example

Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  3,  1;
  1,  8,  9,  4,  1;
  1, 16, 27, 16,  5,  1;
  1, 32, 81, 64, 25,  6,  1;
  ...
From Gus Wiseman, May 01 2021: (Start)
The rows of the triangle are obtained by reading antidiagonals upward in the following table of A(k,n) = n^k, with offset k = 0, n = 1:
         n=1:     n=2:     n=3:     n=4:     n=5:     n=6:
   k=0:   1        1        1        1        1        1
   k=1:   1        2        3        4        5        6
   k=2:   1        4        9       16       25       36
   k=3:   1        8       27       64      125      216
   k=4:   1       16       81      256      625     1296
   k=5:   1       32      243     1024     3125     7776
   k=6:   1       64      729     4096    15625    46656
   k=7:   1      128     2187    16384    78125   279936
   k=8:   1      256     6561    65536   390625  1679616
   k=9:   1      512    19683   262144  1953125 10077696
  k=10:   1     1024    59049  1048576  9765625 60466176
(End)

Maple

E := (n, x) -> `if`(n=0, 1, x*(1-x)*diff(E(n-1, x), x)+E(n-1, x)*(1+(n-1)*x));
G := (n, x) -> E(n, x)/(1-x)^(n+1);
A009998 := (n, k) -> coeff(series(G(n-k, x), x, 18), x, k);
seq(print(seq(A009998(n, k), k=0..n)), n=0..6);
# Peter Luschny, Aug 02 2010

Mathematica

Flatten[Table[(j+1)^(i-j), {i, 0, 20}, {j, 0, i}]] (* Harvey P. Dale, Dec 25 2012 *)

Prog

(Haskell)
a009998 n k = (k + 1) ^ (n - k)
a009998_row n = a009998_tabl !! n
a009998_tabl = map reverse a009999_tabl
-- Reinhard Zumkeller, Feb 02 2014
(PARI) T(i, j)=(j+1)^(i-j) \\ Charles R Greathouse IV, Feb 06 2017

Crossrefs

Row sums give A026898.

Cf. A088956, A123125, A179927.

Column n = 2 of the array is A000079.

Column n = 3 of the array is A000244.

Row k = 2 of the array is A000290.

Row k = 3 of the array is A000578.

Diagonal n = k of the array is A000312.

Diagonal n = k + 1 of the array is A000169.

Diagonal n = k + 2 of the array is A000272.

The transpose of the array is A009999.

The numbers of divisors of the entries are A343656 (row sums: A343657).

A007318 counts k-sets of elements of {1..n}.

A059481 counts k-multisets of elements of {1..n}.

Cf. A002109, A062319, A066959, A143773, A176029, A204688, A326358, A327527.

Sequence in context: A099239 A167630 A322264 * A113993 A103323 A329332

Adjacent sequences: A009995 A009996 A009997 * A009999 A010000 A010001

Keyword

tabl,nonn,easy,nice

Author

N. J. A. Sloane

Extensions

a(62) corrected to 512 by T. D. Noe, Dec 20 2007

Status

approved