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A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem. 13
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

LINKS

Table of n, a(n) for n=0..77.

L. Euler, De mirabilibus proprietatibus numerorum pentagonalium

L. Euler, On the remarkable properties of the pentagonal numbers

Eric Weisstein's World of Mathematics, Pentagonal Number Theorem

FORMULA

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

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

EXAMPLE

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, ...

...

PROG

(GWbasic)' A program (with two A-numbers) for the square array of the example section.

10 DIM A057077(100), A195152(15, 10), T(15, 10)

20 FOR K = 0 TO 10 'Column 0-10

30 T(0, K) = 1     'Row 0

40 FOR N = 1 TO 15 'Rows 1-15

50 FOR J = 1 TO N

60 IF A195152(J, K) <= N THEN T(N, K) = T(N, K) + A057077(J-1) * T(N - A195152(J, K), K)

70 NEXT J

80 NEXT N

90 NEXT K

100 FOR N = 0 TO 15: FOR K = 0 TO 10

110 PRINT T(N, K);

120 NEXT K: PRINT: NEXT N

130 END

CROSSREFS

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

For another version see A195825.

Cf. A057077, A195152.

Sequence in context: A323873 A099020 A179438 * A089688 A229706 A319421

Adjacent sequences:  A211967 A211968 A211969 * A211971 A211972 A211973

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, Jun 10 2012

STATUS

approved

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Last modified February 19 19:06 EST 2020. Contains 332047 sequences. (Running on oeis4.)