A155865 Triangle 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, read by rows.

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

Offset

0,8

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = coefficients of (p(n, x)), where p(n, x) = 1 + x^n + x*((d/dx) (x+1)^n) and T(0, 0) = 1.

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

Sum_{k=0..n} T(n, k) = 2 + A001787(n-1) - (3/4)*[n==0]. - R. J. Mathar, Jul 17 2012

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

T(n, k) = (n-1)*binomial(n-2, k-1) with T(n, 0) = T(n, n) = 1.

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)

T(2*n, n) = A002457(n). - Alois P. Heinz, Dec 05 2018

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, 1 + x^n + 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<2, 1, Product[(i-1)^q, {i, 2, n}]];
T[n_, m_, q_]= c[n, q]/(c[m, q]*c[n-m, q]);
Flatten[Table[T[n, m, q], {n, 0, 12}, {m, 0, n}]] (* 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 */
(Magma)
A155865:= func< n, k | k eq 0 or k eq n select 1 else (n-1)*Binomial(n-2, k-1) >;
[A155865(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 04 2021
(Sage)
def A155865(n, k): return 1 if (k==0 or k==n) else (n-1)*binomial(n-2, k-1)
flatten([[A155865(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

Crossrefs

Cf. A002457 (T(2*n, n)), A155863, A155864.

Sequence in context: A131791 A308497 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