login
A080251
Paired decomposition of tetrahedral numbers A000292 arranged as number triangle.
2
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, 14, 14, 18, 18, 20, 20, 9, 9, 16, 16, 21, 21, 24, 24, 25, 10, 10, 18, 18, 24, 24, 28, 28, 30, 30, 11, 11, 20, 20, 27, 27, 32, 32, 35, 35, 36
OFFSET
0,2
COMMENTS
Used in construction of Stirling-like number triangle A080416.
LINKS
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv preprint arXiv:1302.4694 [math.CO], 2013.
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.
FORMULA
T(n,k) = [k<=n]*floor((k+2)/2)*(n-k+floor((k+3)/2)). - Paul Barry, Jun 14 2010
Also generated by the product of pairs of integers 0 <= r1,r2 <= n whose sum is n+2.
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
EXAMPLE
Rows are
1;
2, 2;
3, 3, 4;
4, 4, 6, 6;
5, 5, 8, 8, 9;
...
Row sums are 1, 4, 10, 20, ... or C(n+3,3) = A000292(n-1).
MATHEMATICA
T[n_, k_] := If[EvenQ[k], (k+2)(2n-k+2)/4, (k+1)(2n-k+3)/4];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 13 2018 *)
T[ n_, k_] := -(Floor(k/2) + 1)*(Floor(k/2) - n - 1); (* Michael Somos, Oct 12 2022 *)
PROG
(PARI) {T(n, k) = -(k\2 + 1)*(k\2 - n - 1)}; /* Michael Somos, Oct 12 2022 */
CROSSREFS
Sequence in context: A103358 A063084 A127079 * A220032 A219773 A187446
KEYWORD
nonn,tabl
AUTHOR
Paul Barry, Feb 17 2003
EXTENSIONS
Edited by Ken Joffaniel M Gonzales, Jul 04 2010
STATUS
approved