A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 3, 1, 1, 1, 10, 5, 2, 1, 1, 1, 16, 7, 3, 1, 1, 1, 1, 24, 11, 4, 2, 1, 1, 1, 1, 36, 15, 5, 3, 1, 1, 1, 1, 1, 54, 22, 7, 4, 2, 1, 1, 1, 1, 1, 78, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 112, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1

Offset: 0

Author

Omar E. Pol, Jun 10 2012

Keywords

• nonn
• tabl

Comments

In the infinite square array if k is positive then column k is related to the generalized m-gonal numbers, where m = k+4. For example: column 1 is related to the generalized pentagonal numbers A001318. Column 2 is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on...

In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041. It seems unusual that the partition numbers are located in a middle column (see below row 1 of the table):

========================================================

. Column k of

. this square

. Generalized Triangle Triangle array A211970

k m m-gonal "A" "B" [row sums of

. numbers triangle "B"

. (if k>=1) with a(0)=1,

. if k >= 0]

========================================================

0 4 A008794 - - A211971

1 5 A001318 A195310 A175003 A000041

2 6 A000217 A195826 A195836 A006950

3 7 A085787 A195827 A195837 A036820

4 8 A001082 A195828 A195838 A195848

5 9 A118277 A195829 A195839 A195849

6 10 A074377 A195830 A195840 A195850

7 11 A195160 A195831 A195841 A195851

8 12 A195162 A195832 A195842 A195852

9 13 A195313 A195833 A195843 A196933

10 14 A195818 A210944 A210954 A210964

...

It appears that column 2 of the square array is A006950.

It appears that column 3 of the square array is A036820.

The partial sums of column 0 give A015128. - Omar E. Pol, Feb 09 2014

Examples

			Array begins:
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
2,     2,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
4,     3,   2,   1,   1,   1,  1,  1,  1,  1,  1, ...
6,     5,   3,   2,   1,   1,  1,  1,  1,  1,  1, ...
10,    7,   4,   3,   2,   1,  1,  1,  1,  1,  1, ...
16,   11,   5,   4,   3,   2,  1,  1,  1,  1,  1, ...
24,   15,   7,   4,   4,   3,  2,  1,  1,  1,  1, ...
36,   22,  10,   5,   4,   4,  3,  2,  1,  1,  1, ...
54,   30,  13,   7,   4,   4,  4,  3,  2,  1,  1, ...
78,   42,  16,  10,   5,   4,  4,  4,  3,  2,  1, ...
112,  56,  21,  12,   7,   4,  4,  4,  4,  3,  2, ...
160,  77,  28,  14,  10,   5,  4,  4,  4,  4,  3, ...
224, 101,  35,  16,  12,   7,  4,  4,  4,  4,  4, ...
312, 135,  43,  21,  13,  10,  5,  4,  4,  4,  4, ...
432, 176,  55,  27,  14,  12,  7,  4,  4,  4,  4, ...
...
		

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

Crossrefs

Columns (0-10): A211971, A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.

For another version see A195825.

Cf. A057077, A195152.

Formula

T(n,k) = A211971(n), if k = 0.

T(n,k) = A195825(n,k), if k >= 1.