A195310 Triangle read by rows with T(n,k) = n - A001318(k), n >= 1, k >= 1, if (n - A001318(k)) >= 0.

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, 15, 14, 11, 9, 4, 1, 16, 15, 12, 10, 5, 2, 17, 16, 13, 11, 6, 3, 18, 17, 14, 12, 7, 4

Offset: 1

Author

Omar E. Pol, Sep 21 2011

Keywords

• nonn
• tabf

Comments

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A001318(k). This sequence is related to Euler's Pentagonal Number Theorem. A000041(a(n)) gives the absolute value of A175003(n). To get the number of partitions of n see the example.

Examples

			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.
		

Links

• L. Euler, De mirabilibus proprietatibus numerorum pentagonalium
• 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

Crossrefs

Row sums give A195311.

Cf. A000041, A001318, A010815, A026741, A057077, A175003.

Programs

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 *)

Formula

A175003(n,k) = A057077(k-1)*A000041(T(n,k)), n >= 1, k >= 1.

Extensions

Name essentially suggested by Franklin T. Adams-Watters (see history), Sep 21 2011