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
KEYWORD
AUTHOR
Roger L. Bagula, Jan 29 2009
EXTENSIONS
Edited and name clarified by Franck Maminirina Ramaharo, Dec 04 2018
STATUS
approved