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A125234 Triangle T(n,k) read by rows: the k-th column contains the k-fold iterated partial sum of A000566.

%I A125234 #15 Oct 26 2023 10:20:14
%S A125234 1,7,1,18,8,1,34,26,9,1,55,60,35,10,1,81,115,95,45,11,1,112,196,210,
%T A125234 140,56,12,1,148,308,406,350,196,68,13,1,189,456,714,756,546,264,81,
%U A125234 14,1,235,645,1170,1470,1302,810,345,95,15,1,286,880,1815,2640,2772,2112,1155,440,110,16,1
%N A125234 Triangle T(n,k) read by rows: the k-th column contains the k-fold iterated partial sum of A000566.
%C A125234 The leftmost column contains the heptagonal numbers A000566.
%C A125234 The adjacent columns to the right are A002413, A002418, A027800, A051946, A050484.
%C A125234 Row sums = 1, 8, 27, 70, 161, 348, 727, ... = 6*(2^n-1)-5*n.
%D A125234 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, 1966, p. 189.
%F A125234 T(n,0) = A000566(n). T(n,k) = T(n-1,k) + T(n-1,k-1), k>0.
%e A125234 First few rows of the triangle are:
%e A125234   1;
%e A125234   7, 1;
%e A125234   18, 8, 1;
%e A125234   34, 26, 9, 1;
%e A125234   55, 60, 35, 10, 1;
%e A125234   81, 115, 95, 45, 11, 1;
%e A125234   112, 196, 210, 140, 56, 12, 1;
%e A125234 Example: T(6,2) = 95 = 35 + 60 = T(5,2) + T(5,1).
%p A125234 A000566 := proc(n) n*(5*n-3)/2 ; end: A125234 := proc(n,k) if k = 0 then A000566(n); elif k>= n then 0 ; else procname(n-1,k-1)+procname(n-1,k) ; fi; end: seq(seq(A125234(n,k),k=0..n-1),n=1..16) ; # _R. J. Mathar_, Sep 09 2009
%t A125234 A000566[n_] := n(5n-3)/2;
%t A125234 T[n_, k_] := Which[k == 0, A000566[n], k >= n, 0, True, T[n-1, k-1] + T[n-1, k] ];
%t A125234 Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 11}] // Flatten (* _Jean-François Alcover_, Oct 26 2023, after _R. J. Mathar_ *)
%Y A125234 Cf. A000566, A002413, A002418, A027800, A051946, A050484.
%Y A125234 Analogous triangles for the hexagonal and pentagonal numbers are A125233 and A125232.
%K A125234 nonn,tabl
%O A125234 1,2
%A A125234 _Gary W. Adamson_, Nov 24 2006
%E A125234 Edited and extended by _R. J. Mathar_, Sep 09 2009