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A009998 Triangle in which j-th entry in i-th row is (j+1)^(i-j). 16
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 (list; table; graph; refs; listen; history; text; internal format)
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 - Peter Luschny, Aug 02 2010

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;

  ...

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, A009999 (mirrored).

Sequence in context: A099239 A167630 A322264 * A113993 A103323 A092056

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

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Last modified October 23 14:45 EDT 2019. Contains 328345 sequences. (Running on oeis4.)