cp's OEIS Frontend

A204842 Triangle by rows relating to A081696.

%I A204842 #11 Jun 02 2025 07:37:59
%S A204842 1,1,2,3,4,2,9,12,6,2,29,38,20,8,2,97,126,68,30,10,2,333,430,236,110,
%T A204842 42,12,2,1165,1498,832,402,166,56,14,2,4135,5300,2970,1472,640,238,72,
%U A204842 16,2,14845,18980,10710,5410,2440,968,328,90,18,2
%N A204842 Triangle by rows relating to A081696.
%F A204842 n-th row of the triangle is the top row of M^n, where M is the following infinite square production matrix:
%F A204842 1, 2, 0, 0, 0, 0,...
%F A204842 1, 1, 1, 0, 0, 0,...
%F A204842 1, 1, 1, 1, 0, 0,...
%F A204842 1, 1, 1, 1, 1, 0,...
%F A204842 1, 1, 1, 1, 1, 1,...
%F A204842 ...
%F A204842 Conjecture: T(n,1) = 2*A109262(n). T(n,2)=2*A109263(n). - _R. J. Mathar_, Jul 21 2015
%e A204842 First few rows of the triangle =
%e A204842 1;
%e A204842 1, 2;
%e A204842 3, 4, 2;
%e A204842 9, 12, 6, 2;
%e A204842 29, 38, 20, 8, 2;
%e A204842 97, 126, 68, 30, 10, 2;
%e A204842 333, 430, 236, 110, 42, 12, 2;
%e A204842 1165, 1498, 832, 402, 166, 56, 14, 2;
%e A204842 4135, 5300, 2970, 1472, 640, 238, 72, 16, 2;
%e A204842 14845, 18980, 10710, 5410, 2440, 968, 328, 90, 18, 2;
%e A204842 ...
%e A204842 Top row of M^3 = [9, 12, 6, 2, 0, 0, 0,...]
%p A204842 A204842T := proc(n,k)
%p A204842     if n =0 and k =1 then
%p A204842         2;
%p A204842     elif k <0 or k >n+1 then
%p A204842         0;
%p A204842     else
%p A204842         1;
%p A204842     end if ;
%p A204842 end proc:
%p A204842 A204842 := proc(n,k)
%p A204842     local A;
%p A204842     A := Matrix(n+1,n+1) ;
%p A204842     for row from 1 to n+1 do
%p A204842     for col from 1 to n+1 do
%p A204842             A[row, col] := A204842T(row-1,col-1) ;
%p A204842     end do:
%p A204842     end do:
%p A204842     Mn := LinearAlgebra[MatrixPower](A , n);
%p A204842     Mn[1,k+1] ;
%p A204842 end proc:
%p A204842 for n from 0 to 10 do
%p A204842     for k from 0 to n do
%p A204842     printf("%d ",A204842(n,k)) ;
%p A204842     end do:
%p A204842     printf("\n") ;
%p A204842 end do:
%Y A204842 Cf. A081696 (first column and also row sums).
%K A204842 nonn,tabl
%O A204842 0,3
%A A204842 _Gary W. Adamson_, Jan 19 2012