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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

%I

%S 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,

%T 60,60,30,6,1,1,7,42,105,140,105,42,7,1,1,8,56,168,280,280,168,56,8,1,

%U 1,9,72,252,504,630,504,252,72,9,1,1,10,90,360,840,1260,1260,840,360,90,10,1

%N 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.

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

%F 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).

%F 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

%F 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

%F From _Franck Maminirina Ramaharo_, Dec 05 2018: (Start)

%F 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)).

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

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

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 2, 2, 1;

%e 1, 3, 6, 3, 1;

%e 1, 4, 12, 12, 4, 1;

%e 1, 5, 20, 30, 20, 5, 1;

%e 1, 6, 30, 60, 60, 30, 6, 1;

%e 1, 7, 42, 105, 140, 105, 42, 7, 1;

%e 1, 8, 56, 168, 280, 280, 168, 56, 8, 1;

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

%e ...

%e ConvOffs transform of (1, 1, 2, 3) = integers of row 4: (1, 3, 6, 3, 1). _Gary W. Adamson_, Jul 09 2012

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

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

%t (* or *)

%t q = 1;

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

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

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

%o (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 */

%Y Cf. A155863, A155864.

%Y T(2n,n) gives A002457.

%K nonn,tabl,easy

%O 0,8

%A _Roger L. Bagula_, Jan 29 2009

%E Edited and name clarified by _Franck Maminirina Ramaharo_, Dec 04 2018

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Last modified March 31 16:23 EDT 2020. Contains 333151 sequences. (Running on oeis4.)