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%I #19 Sep 08 2022 08:45:41
%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 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.
%H G. C. Greubel, <a href="/A155865/b155865.txt">Rows n = 0..50 of the triangle, flattened</a>
%F 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.
%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 Sum_{k=0..n} T(n, k) = 2 + A001787(n-1) - (3/4)*[n==0]. - _R. J. Mathar_, Jul 17 2012
%F From _Franck Maminirina Ramaharo_, Dec 05 2018: (Start)
%F T(n, k) = (n-1)*binomial(n-2, k-1) with T(n, 0) = T(n, n) = 1.
%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)
%F T(2*n, n) = A002457(n). - _Alois P. Heinz_, Dec 05 2018
%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, 1 + x^n + 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<2, 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[T[n, m, q], {n,0,12}, {m, 0, n}]] (* _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 */
%o (Magma)
%o A155865:= func< n,k | k eq 0 or k eq n select 1 else (n-1)*Binomial(n-2, k-1) >;
%o [A155865(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 04 2021
%o (Sage)
%o def A155865(n,k): return 1 if (k==0 or k==n) else (n-1)*binomial(n-2, k-1)
%o flatten([[A155865(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 04 2021
%Y Cf. A002457 (T(2*n, n)), A155863, A155864.
%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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