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A080251 Paired decomposition of tetrahedral numbers A000292 arranged as number triangle.

%I A080251 #43 Oct 13 2022 12:51:39
%S A080251 1,2,2,3,3,4,4,4,6,6,5,5,8,8,9,6,6,10,10,12,12,7,7,12,12,15,15,16,8,8,
%T A080251 14,14,18,18,20,20,9,9,16,16,21,21,24,24,25,10,10,18,18,24,24,28,28,
%U A080251 30,30,11,11,20,20,27,27,32,32,35,35,36
%N A080251 Paired decomposition of tetrahedral numbers A000292 arranged as number triangle.
%C A080251 Used in construction of Stirling-like number triangle A080416.
%H A080251 R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, <a href="http://arxiv.org/abs/1302.4694">Dually weighted Stirling-type sequences</a>, arXiv preprint arXiv:1302.4694 [math.CO], 2013.
%H A080251 R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, <a href="http://dx.doi.org/10.1016/j.ejc.2014.06.010">Dually weighted Stirling-type sequences</a>, Europ. J. Combin., 43, 2015, 55-67.
%F A080251 T(n,k) = [k<=n]*floor((k+2)/2)*(n-k+floor((k+3)/2)). - _Paul Barry_, Jun 14 2010
%F A080251 Also generated by the product of pairs of integers 0 <= r1,r2 <= n whose sum is n+2.
%F A080251 Viewed as a square array: T(n,2*k) = k*(k+n); T(n,2*k+1) = (k+1)*(k+n). - _Luc Rousseau_, Dec 11 2017
%e A080251 Rows are
%e A080251   1;
%e A080251   2, 2;
%e A080251   3, 3, 4;
%e A080251   4, 4, 6, 6;
%e A080251   5, 5, 8, 8, 9;
%e A080251   ...
%e A080251 Row sums are 1, 4, 10, 20, ... or C(n+3,3) = A000292(n-1).
%t A080251 T[n_, k_] := If[EvenQ[k], (k+2)(2n-k+2)/4, (k+1)(2n-k+3)/4];
%t A080251 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 13 2018 *)
%t A080251 T[ n_, k_] := -(Floor(k/2) + 1)*(Floor(k/2) - n - 1); (* _Michael Somos_, Oct 12 2022 *)
%o A080251 (PARI) {T(n, k) = -(k\2 + 1)*(k\2 - n - 1)}; /* _Michael Somos_, Oct 12 2022 */
%K A080251 nonn,tabl
%O A080251 0,2
%A A080251 _Paul Barry_, Feb 17 2003
%E A080251 Edited by _Ken Joffaniel M Gonzales_, Jul 04 2010