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%I #14 Aug 17 2022 16:21:07
%S 1,1,1,1,3,1,1,3,5,1,1,3,8,7,1,1,3,8,16,9,1,1,3,8,20,27,11,1,1,3,8,20,
%T 43,41,13,1,1,3,8,20,48,81,58,15,1,1,3,8,20,48,106,138,78,17,1,1,3,8,
%U 20,48,112,213,218,101,19,1
%N Array read by antidiagonals: the binomial transform of the sequence (1,2,..n,0,0,0..) in row n.
%C Create an array by rows: (binomial transforms of 1,0,0,0,...; 1,2,0,0,0,...; 1,2,3,0,0,0,...; etc.). Antidiagonals of the array become rows of the triangle.
%e First few rows of the array:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 3, 5, 7, 9, 11, 13, 15, 17, ...
%e 1, 3, 8, 16, 27, 41, 58, 78, 101, ... A104249
%e 1, 3, 8, 20, 43, 81, 138, 218, ... A139488
%e 1, 3, 8, 20, 48, 106, 213, ...
%e 1, 3, 8, 20, 48, 112, 249, ...
%e ...
%e Diagonals converge to A001792, binomial transform of (1,2,3,...); and the first few rows of the triangle created by reading upwards antidiagonals are:
%e 1
%e 1, 1;
%e 1, 3, 1;
%e 1, 3, 5, 1;
%e 1, 3, 8, 7, 1;
%e 1, 3, 8, 16, 9, 1;
%e 1, 3, 8, 20, 27, 22, 1;
%e ...
%e a(4), a(5), a(6) = 1, 3, 1 = antidiagonals of the array becoming row three of the triangle.
%p A116445 := proc(n,k)
%p local a,i ;
%p a := 0 ;
%p for i from 0 to n do
%p a := a+binomial(k,i)*(i+1) ;
%p end do:
%p a ;
%p end proc:
%p seq(seq(A116445(d-k,k),k=0..d),d=0..12) ; # _R. J. Mathar_, Aug 17 2022
%Y Cf. A001629 (antidiagonal sums), A104249.
%K nonn,tabl,easy
%O 1,5
%A _Gary W. Adamson_, Feb 15 2006
%E Detailed NAME by _R. J. Mathar_, Aug 17 2022