OFFSET
1,4
COMMENTS
LINKS
L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem
EXAMPLE
Written as a triangle:
0;
1, 0;
2, 1;
3, 2;
4, 3, 0;
5, 4, 1;
6, 5, 2, 0;
7, 6, 3, 1;
8, 7, 4, 2;
9, 8, 5, 3;
10, 9, 6, 4;
11, 10, 7, 5, 0;
12, 11, 8, 6, 1;
13, 12, 9, 7, 2;
14, 13, 10, 8, 3, 0;
.
For n = 15, consider row 15 which lists the numbers 14, 13, 10, 8, 3, 0. From Euler's Pentagonal Number Theorem we have that the number of partitions of 15 is p(15) = p(14) + p(13) - p(10) - p(8) + p(3) + p(0) = 135 + 101 - 42 - 22 + 3 + 1 = 176.
MATHEMATICA
rows = 20;
a1318[n_] := If[EvenQ[n], n(3n/2+1)/4, (n+1)(3n+1)/8];
T[n_, k_] := n - a1318[k];
Table[DeleteCases[Table[T[n, k], {k, 1, n}], _?Negative], {n, 1, rows}] // Flatten (* Jean-François Alcover, Sep 22 2018 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Sep 21 2011
EXTENSIONS
Name essentially suggested by Franklin T. Adams-Watters (see history), Sep 21 2011
STATUS
approved