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A181996 Triangle of Ward numbers T(n,k) (n>=0, k=0 if n=0, otherwise 0 <= k <= n-1) read by rows. 2
1, 1, 3, 1, 15, 10, 1, 105, 105, 25, 1, 945, 1260, 490, 56, 1, 10395, 17325, 9450, 1918, 119, 1, 135135, 270270, 190575, 56980, 6825, 246, 1, 2027025, 4729725, 4099095, 1636635, 302995, 22935, 501, 1, 34459425, 91891800, 94594500, 47507460, 12122110, 1487200 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

It appears that the sum of row(n) is A000311(n+1). - Michel Marcus, Feb 07 2013

Conjecture on row sums was proved in the first paragraph of the formula section of the reverse matrix A134991 in 2008 (e.g.f. evaluated at t=1). - Tom Copeland, Jan 03 2016

REFERENCES

Charles Jordan, Calculus of Finite Differences, Chelsea 1950, p. 172, Table C_{m, i}.

LINKS

Table of n, a(n) for n=0..42.

L. Carlitz, The coefficients in an asymptotic expansion and certain related numbers, Duke Math. J., vol 35 (1968), p. 83-90. See page 85.

José L. Cereceda, Figurate numbers and sums of powers of integers, arXiv:2001.03208 [math.NT], 2020. See Table 8 p. 11.

Lane Clark, Asymptotic normality of the Ward numbers, Discrete Math. 203 (1999), no. 1-3, 41-48. MR1696232 (2000d:11101)

T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms

Andrew Elvey Price, Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.

EXAMPLE

Triangle begins:

      1

      1

      3     1

     15    10    1

    105   105   25    1

    945  1260  490   56   1

  10395 17325 9450 1918 119 1 ...

PROG

(PARI) T(n, k) = {if (n == 0, return(1)); if (k == 0, return (prod(x=2, n, 2*x-1))); if (k == n, return (0)); return((2*n-1-k)*T(n-1, k) + (n-k)*T(n-1, k-1)); } \\ Michel Marcus, Feb 07 2013

CROSSREFS

See A134991, which is the mirror image and is the main entry for this triangle, for further information.

Cf. A000311.

Sequence in context: A147020 A134685 A130757 * A014621 A144006 A193966

Adjacent sequences:  A181993 A181994 A181995 * A181997 A181998 A181999

KEYWORD

nonn,tabf

AUTHOR

N. J. A. Sloane, Apr 05 2012

EXTENSIONS

More terms from Michel Marcus, Feb 07 2013

STATUS

approved

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Last modified June 7 02:48 EDT 2020. Contains 334836 sequences. (Running on oeis4.)