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
Examples
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
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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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
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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 *) -
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 */
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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
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
Extensions
Edited and name clarified by Franck Maminirina Ramaharo, Dec 04 2018