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A125105 Triangular array with the first half of the odd-indexed rows of A048004.

%I A125105 #21 Jan 27 2024 10:34:17
%S A125105 1,1,4,1,12,11,1,33,47,27,1,88,185,127,63,1,232,694,563,303,143,1,609,
%T A125105 2526,2400,1394,687,319,1,1596,9012,9960,6215,3186,1519,703,1,4180,
%U A125105 31709,40534,27095,14401,7026,3311,1535,1,10945,110469,162538,116143,63872,31808,15218,7151,3327
%N A125105 Triangular array with the first half of the odd-indexed rows of A048004.
%C A125105 A000079 counts compositions admitting a variety of triangular views; for example, A048004 and A105147. The subtable formed from the odd rows of A048004 has row sums 1, 8, 44, 208, 912, ... . Because only the first half of rows of A048004 is transferred to this triangle here, there is a difference between row sums of A048004 and row sums here, A045623(n-1).
%F A125105 T(n,k) = A048004(2*n-1,k), 0 <= k < n. - _R. J. Mathar_, Nov 23 2007
%e A125105 The odd-indexed rows of triangle A048004 begin
%e A125105   1  1
%e A125105   1  4  2 1
%e A125105   1 12 11 5 2 1
%e A125105   ...
%e A125105 so the triangle here begins
%e A125105   1
%e A125105   1  4
%e A125105   1 12 11
%e A125105   ...
%p A125105 A048004 := proc(n,k) option remember ; if k < 0 or k > n then 0; elif k = 0 or k = n then 1; else 2*procname(n-1,k)+procname(n-1,k-1)-2*procname(n-2,k-1)+procname(n-k-1,k-1)-procname(n-k-2,k) ; fi ; end:
%p A125105 A125105 := proc(n,k) A048004(2*n-1,k) ; end:
%p A125105 for n from 1 to 13 do for k from 0 to n-1 do printf("%d ",A125105(n,k)) ; od: od: # _R. J. Mathar_, Nov 23 2007
%t A125105 B[n_, k_] := B[n, k] = If[n == 0 || k == 1, 1, Sum[B[n - j, k], {j, 1, Min[n, k]}]];
%t A125105 A048004[n_, k_] := B[n + 1, k + 1] - B[n + 1, k];
%t A125105 T[n_, k_] := A048004[2 n - 1, k];
%t A125105 Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* _Jean-François Alcover_, Jan 27 2024, after Maple code here and in A048004 *)
%Y A125105 Cf. A008464 (row sums), A008464, A045623, A048004, A105147, A125104.
%K A125105 easy,nonn,tabl
%O A125105 1,3
%A A125105 _Alford Arnold_, Dec 07 2006
%E A125105 More terms from _R. J. Mathar_, Nov 23 2007