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A124727 Triangle read by rows: T(n,k)=k*binomial(n-1,k-1)+binomial(n-1,k) (1<=k<=n).

%I A124727 #9 Nov 21 2013 12:49:01
%S A124727 1,2,2,3,5,3,4,9,10,4,5,14,22,17,5,6,20,40,45,26,6,7,27,65,95,81,37,7,
%T A124727 8,35,98,175,196,133,50,8,9,44,140,294,406,364,204,65,9,10,54,192,462,
%U A124727 756,840,624,297,82,10,11,65,255,690,1302,1722,1590,1005,415,101,11,12,77
%N A124727 Triangle read by rows: T(n,k)=k*binomial(n-1,k-1)+binomial(n-1,k) (1<=k<=n).
%C A124727 Triangle is P*M, where P is Pascal's triangle as an infinite lower triangular matrix and M is the infinite bidiagonal matrix with (1,2,3...) in the main diagonal and (1,1,1...) in the subdiagonal.
%e A124727 First few rows of the triangle are:
%e A124727 1;
%e A124727 2, 2;
%e A124727 3, 5, 3;
%e A124727 4, 9, 10, 4;
%e A124727 5, 14, 22, 17, 5;
%e A124727 6, 20, 40, 45, 26, 6
%e A124727 ...
%p A124727 T:=(n,k)->k*binomial(n-1,k-1)+binomial(n-1,k): for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%t A124727 Flatten[Table[k Binomial[n-1,k-1]+Binomial[n-1,k],{n,20},{k,n}]] (* _Harvey P. Dale_, Jan 28 2012 *)
%Y A124727 Row sums = A047859: (1, 4, 11, 27, 143, 319...) A124726 is generated in an analogous manner by taking M*P instead of P*M.
%K A124727 nonn,tabl
%O A124727 1,2
%A A124727 _Gary W. Adamson_ & _Roger L. Bagula_, Nov 05 2006
%E A124727 Edited by _N. J. A. Sloane_, Nov 24 2006