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A123097 Triangle read by rows: T(n,k) = binomial(n-2, k-1) + n*binomial(n-1, k-1), 1 <= k <= n, starting with T(1, 1) = 1.

%I A123097 #21 Sep 08 2022 08:45:28
%S A123097 1,3,2,4,7,3,5,14,13,4,6,23,33,21,5,7,34,66,64,31,6,8,47,115,150,110,
%T A123097 43,7,9,62,183,300,295,174,57,8,10,79,273,539,665,525,259,73,9,11,98,
%U A123097 388,896,1330,1316,868,368,91,10,12,119,531,1404,2436,2898,2394,1356,504,111,11
%N A123097 Triangle read by rows: T(n,k) = binomial(n-2, k-1) + n*binomial(n-1, k-1), 1 <= k <= n, starting with T(1, 1) = 1.
%C A123097 Triangle is M*P, where M is the infinite bidiagonal matrix with (1,2,3,...) in the main diagonal and (1,1,1,...) in the subdiagonal and P is Pascal's triangle as an infinite lower triangular matrix. The triangle A124727 is P*M.
%H A123097 G. C. Greubel, <a href="/A123097/b123097.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A123097 Sum_{k=1..n} T(n, k) = 2^(n-2)*(2*n + 1) - (1/2)*[n=1] = A052951(n-1). - _G. C. Greubel_, Jul 21 2021
%e A123097 First few rows of the triangle are
%e A123097   1;
%e A123097   3,  2;
%e A123097   4,  7,  3;
%e A123097   5, 14, 13,  4
%e A123097   6, 23, 33, 21,  5;
%e A123097   7, 34, 66, 64, 31,  6;
%e A123097   ...
%p A123097 T:=proc(n,k) if n=1 and k=1 then 1 elif n=1 then 0 else binomial(n-2,k-1)+n*binomial(n-1,k-1) fi end: for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%t A123097 T[n_, k_]= If[n==1, 1, Binomial[n-2, k-1] + n*Binomial[n-1, k-1]];
%t A123097 Table[T[n, k], {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Jul 21 2021 *)
%o A123097 (PARI) T(n,k) = if ((n==1), (k==1), binomial(n-2,k-1)+n*binomial(n-1,k-1));
%o A123097 matrix(11, 11, n, k, T(n,k)) \\ _Michel Marcus_, Nov 09 2019
%o A123097 (Magma)
%o A123097 A123097:= func< n,k | n eq 1 select 1 else Binomial(n-2, k-1) + n*Binomial(n-1, k-1) >;
%o A123097 [A123097(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 21 2021
%o A123097 (Sage)
%o A123097 def A123097(n,k): return 1 if (n==1) else binomial(n-2, k-1) + n*binomial(n-1, k-1)
%o A123097 flatten([[A123097(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Jul 21 2021
%Y A123097 Cf. A052951 (row sums).
%K A123097 nonn,tabl
%O A123097 1,2
%A A123097 _Gary W. Adamson_ and _Roger L. Bagula_, Nov 05 2006
%E A123097 Edited by _N. J. A. Sloane_, Nov 24 2006