A104746 - OEIS

%I #17 Mar 30 2024 07:53:17

%S 1,1,3,1,4,7,1,5,12,15,1,6,17,32,31,1,7,22,49,80,63,1,8,27,66,129,192,

%T 127,1,9,32,83,178,321,448,255,1,10,37,100,227,450,769,1024,511,1,11,

%U 42,117,276,579,1090,1793,2304,1023,1,12,47,134,325,708,1411,2562,4097,5120,2047,1,13,52,151,374,837,1732,3331,5890,9217,11264,4095

%N Array T(n,k) read by antidiagonals: T(1,k) = 2^k-1 and recursively T(n,k) = T(n-1,k) + A000337(k-1), n,k >= 1.

%C Generally, row n of the array is the binomial transform for 0, 1, n, 2n-1, 3n-2, 4n-3, ...

%F T(2,k) = A001787(k), binomial transform of 0, 1, 2, 3, 4, 5, 6, ...

%F T(3,k) = A000337(k), binomial transform of 0, 1, 3, 5, 7, 9, 11, ...

%F T(4,k) = A027992(k-1), binomial transform of 0, 1, 4, 7, 10, 13, 16, 19, 22, 25, ...

%F T(5,k) = binomial transform of 0, 1, 5, 9, 13, 17, 21, 25, 29, ...

%e To the first row, add the terms 0, 1, 5, 17, 49, 129, ... as indicated:

%e   1, 3,  7, 15, 31,  63, ...

%e   0, 1,  5, 17, 49, 129, ... (getting row 2 of the array:

%e   1, 4, 12, 32, 80, 192, ... (= A001787, binomial transform for 1,2,3, ...)

%e Repeat the operation, getting the following array T(n,k):

%e   1, 3,  7, 15,  31,  63, ...

%e   1, 4, 12, 32,  80, 192, ...

%e   1, 5, 17, 49, 129, 321, ...

%e   1, 6, 22, 66, 178, 450, ...

%p A000337 := proc(n)

%p         1+(n-1)*2^n ;

%p end proc:

%p A104746 := proc(n,k)

%p         option remember;

%p         if n=  1 then

%p                 2^k-1 ;

%p         else

%p                 procname(n-1,k)+A000337(k-1) ;

%p         end if;

%p end proc:

%p for d from 1 to 12 do

%p         for k from 1 to d do

%p                 n := d-k+1 ;

%p                 printf("%d,",A104746(n,k)) ;

%p         end do:

%p end do; # _R. J. Mathar_, Oct 30 2011

%t A000337[n_] := (n - 1)*2^n + 1;

%t T[1, k_] := 2^k - 1;

%t T[n_, k_] := T[n, k] = T[n - 1, k] + A000337[k - 1];

%t Table[T[n - k + 1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 30 2024 *)

%Y Cf. A104747 (antidiagonal sums), A001787, A000337, A027992, A059823.

%K nonn,tabl

%O 1,3

%A _Gary W. Adamson_, Mar 23 2005

%E Terms corrected by _R. J. Mathar_, Oct 30 2011