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A155863 T(n,k) = n*(n^2 - 1)*binomial(n - 2, k - 1) for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, triangle read by rows. 4
1, 1, 1, 1, 6, 1, 1, 24, 24, 1, 1, 60, 120, 60, 1, 1, 120, 360, 360, 120, 1, 1, 210, 840, 1260, 840, 210, 1, 1, 336, 1680, 3360, 3360, 1680, 336, 1, 1, 504, 3024, 7560, 10080, 7560, 3024, 504, 1, 1, 720, 5040, 15120, 25200, 25200, 15120, 5040, 720, 1, 1, 990, 7920, 27720, 55440, 69300, 55440, 27720, 7920, 990, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Table of n, a(n) for n=0..65.

FORMULA

Row 0 is 1, and row n = coefficients in the expansion of p(x,n) = x^n + 1 + x*((d/dx)^3 (x + 1)^(n + 1)).

From Franck Maminirina Ramaharo, Dec 03 2018: (Start)

n-th row polynomial is x^n + n*(n^2 - 1)*x*(x + 1)^(n - 2) + (1 + (-1)^(2^n))/2.

G.f.: 1/(1 - y) + 1/(1 - x*y) + (6*x*y^2)/(1 - y - x*y)^4 - 1.

E.g.f.: exp(y) + exp(x*y) + (3*x*y^2 + (x + x^2)*y^3)*exp((1 + x)*y) - 1. (End)

EXAMPLE

Triangle begins:

  1;

  1,   1;

  1,   6,    1;

  1,  24,   24,     1;

  1,  60,  120,    60,     1;

  1, 120,  360,   360,   120,     1;

  1, 210,  840,  1260,   840,   210,     1;

  1, 336, 1680,  3360,  3360,  1680,   336,     1;

  1, 504, 3024,  7560, 10080,  7560,  3024,   504,    1,

  1, 720, 5040, 15120, 25200, 25200, 15120,  5040,  720,   1;

  1, 990, 7920, 27720, 55440, 69300, 55440, 27720, 7920, 990, 1;

  ...

MATHEMATICA

p[x_, n_] = If[n == 0, 1, x^n + 1 + x*D[(x + 1)^(n + 1), {x, 3}]];

Flatten[Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}]]

PROG

(Maxima) T(n, k) := ratcoef(expand(x^n + n*(n^2 - 1)*x*(x + 1)^(n - 2) + (1 + (-1)^(2^n))/2), x, k)$

create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 03 2018 */

CROSSREFS

Cf. A155864, A155865

Sequence in context: A060187 A174527 A156139 * A173882 A174045 A169660

Adjacent sequences:  A155860 A155861 A155862 * A155864 A155865 A155866

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Jan 29 2009

EXTENSIONS

Edited and name clarified by Franck Maminirina Ramaharo, Dec 03 2018

STATUS

approved

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Last modified January 16 21:37 EST 2019. Contains 319206 sequences. (Running on oeis4.)