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A125233 Triangle T(n,k) read by rows, the (n-k)-th term of the k times repeated partial sum of the hexagonal numbers, 0 <= k < n, 0 < n.

%I A125233 #22 Sep 14 2023 05:16:41
%S A125233 1,6,1,15,7,1,28,22,8,1,45,50,30,9,1,66,95,80,39,10,1,91,161,175,119,
%T A125233 49,11,1,120,252,336,294,168,60,12,1,153,372,588,630,462,228,72,13,1,
%U A125233 190,525,960,1218,1092,690,300,85,14,1,231,715,1485,2178,2310,1782,990,385,99,15,1
%N A125233 Triangle T(n,k) read by rows, the (n-k)-th term of the k times repeated partial sum of the hexagonal numbers, 0 <= k < n, 0 < n.
%C A125233 Left border = A000384, hexagonal numbers. The following columns are A002412, A002417, A034263, A051947, ...
%C A125233 Row sums = (1, 7, 23, 59, 135, 291, ...) = A126284.
%C A125233 A125232 is the analogous triangle for the pentagonal numbers.
%D A125233 Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1964, p. 189.
%F A125233 T(n,0)=A000384(n). T(n,k) = T(n-1,k) + T(n-1,k-1), k>1. - _R. J. Mathar_, May 03 2008
%e A125233 First few rows of the triangle:
%e A125233    1;
%e A125233    6,   1;
%e A125233   15,   7,   1;
%e A125233   28,  22,   8,   1;
%e A125233   45,  50,  30,   9,  1;
%e A125233   66,  95,  80,  39, 10,  1;
%e A125233   91, 161, 175, 119, 49, 11, 1;
%e A125233   ...
%e A125233 Example: (5,3) = 80 = 30 + 50 = (4,3) + (4,2).
%p A125233 A000384Psum:= proc(n,k) coeftayl( x*(1+3*x)/(1-x)^(3+k),x=0,n) ; end: A125233 := proc(n,k) A000384Psum(n-k,k) ; end: for n from 1 to 15 do for k from 0 to n -1 do printf("%d,",A125233(n,k)) ; od: od: # _R. J. Mathar_, May 03 2008
%t A125233 T[n_, k_] := T[n, k] = Which[k == 0, n (2 n - 1), 1 <= k < n, T[n - 1, k] + T[n - 1, k - 1], True, 0];
%t A125233 Table[T[n, k], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* _Jean-François Alcover_, Sep 14 2023, after _R. J. Mathar_ *)
%Y A125233 Cf. A000384, A002412, A002417, A034263, A051947, A125232.
%K A125233 nonn,tabl
%O A125233 0,2
%A A125233 _Gary W. Adamson_, Nov 24 2006
%E A125233 Edited and extended by _R. J. Mathar_, May 03 2008, and _M. F. Hasler_, Sep 29 2012