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A155865 T(n,k) = (n - 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. 3
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 3, 1, 1, 4, 12, 12, 4, 1, 1, 5, 20, 30, 20, 5, 1, 1, 6, 30, 60, 60, 30, 6, 1, 1, 7, 42, 105, 140, 105, 42, 7, 1, 1, 8, 56, 168, 280, 280, 168, 56, 8, 1, 1, 9, 72, 252, 504, 630, 504, 252, 72, 9, 1, 1, 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

For n >= 1, row n sums to 2 + A001787(n-1).

LINKS

Table of n, a(n) for n=0..77.

FORMULA

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

Define c(n) = Product_{i=2..n} (i - 1), with c(0) = c(1) = 1. Then T(n,m) = c(n)/(c(m)*c(n-m)). - Roger L. Bagula, Mar 09 2010

The triangle is the ConvOffsStoT transform of the natural numbers prefaced with a 1. A row with n integers is the ConvOffs transform of a finite series of the first (n-1) terms in (1, 1, 2, 3, 4, ...).  See A214281 for definitions of the transform. - Gary W. Adamson, Jul 09 2012

From Franck Maminirina Ramaharo, Dec 05 2018: (Start)

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

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

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

EXAMPLE

Triangle begins:

  1;

  1, 1;

  1, 1,  1;

  1, 2,  2,   1;

  1, 3,  6,   3,   1;

  1, 4, 12,  12,   4,   1;

  1, 5, 20,  30,  20,   5,   1;

  1, 6, 30,  60,  60,  30,   6,   1;

  1, 7, 42, 105, 140, 105,  42,   7,  1;

  1, 8, 56, 168, 280, 280, 168,  56,  8, 1;

  1, 9, 72, 252, 504, 630, 504, 252, 72, 9, 1;

  ...

ConvOffs transform of (1, 1, 2, 3) = integers of row 4: (1, 3, 6, 3, 1). Gary W. Adamson, Jul 09 2012

MATHEMATICA

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

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

(* or *)

q = 1;

c[n_, q_] = If[n == 0, 1, If[n == 1, 1, Product[(i - 1)^q, {i, 2, n}]]];

t[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);

Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]] (* Roger L. Bagula, Mar 09 2010 *)

PROG

(Maxima) T(n, k) := if k = 0 or k = n then 1 else (n - 1)*binomial(n - 2, k - 1)$ create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 05 2018 */

CROSSREFS

Cf. A155863, A155864.

T(2n,n) gives A002457.

Sequence in context: A008302 A131791 A010358 * A156133 A010048 A055870

Adjacent sequences:  A155862 A155863 A155864 * A155866 A155867 A155868

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Jan 29 2009

EXTENSIONS

Edited and name clarified by Franck Maminirina Ramaharo, Dec 04 2018

STATUS

approved

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Last modified January 21 15:46 EST 2019. Contains 319350 sequences. (Running on oeis4.)