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A195825 Square array T(n,k) read by antidiagonals, n>=0, k>=1, which arises from a generalization of Euler's Pentagonal Number Theorem. 38
1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 7, 3, 1, 1, 1, 1, 11, 4, 2, 1, 1, 1, 1, 15, 5, 3, 1, 1, 1, 1, 1, 22, 7, 4, 2, 1, 1, 1, 1, 1, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1, 56, 16, 7, 4, 3, 1, 1, 1, 1, 1, 1, 1, 77, 21, 10, 4 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

In the infinite square array the column k is related to the generalized m-gonal numbers, where m = k+4. For example: the first column is related to the generalized pentagonal numbers A001318. The second column is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on... (see the program in which A195152 is a table of generalized m-gonal numbers).

In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041 (see below the first row of the table):

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

.                                          Column k of

.                                          this square

.       Generalized   Triangle  Triangle   array A195825

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

.         numbers                          triangle "B"

.                                          with a(0)=1]

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

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.

Conjecture: if k is odd then column k contains (k+1)/2 plateaus whose levels are the first (k+1)/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5,... 2. Otherwise, if k is even then column k contains k/2 plateaus whose levels are the first k/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5,... 3. The sequence A210843 gives the levels of the plateaus of column k, when k -> infinity. For the visualization of the plateaus see the graph of a column, for example see the graph of A210964. - Omar E. Pol, Jun 21 2012

LINKS

Table of n, a(n) for n=0..81.

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

FORMULA

Column k is asymptotic to exp(Pi*sqrt(2*n/(k+2))) / (8*sin(Pi/(k+2))*n). - Vaclav Kotesovec, Aug 14 2017

EXAMPLE

Array begins:

1,    1,  1,  1,  1,  1,  1,  1,  1,  1,...

1,    1,  1,  1,  1,  1,  1,  1,  1,  1,...

2,    1,  1,  1,  1,  1,  1,  1,  1,  1,...

3,    2,  1,  1,  1,  1,  1,  1,  1,  1,...

5,    3,  2,  1,  1,  1,  1,  1,  1,  1,...

7,    4,  3,  2,  1,  1,  1,  1,  1,  1,...

11,   5,  4,  3,  2,  1,  1,  1,  1,  1,...

15,   7,  4,  4,  3,  2,  1,  1,  1,  1,...

22,  10,  5,  4,  4,  3,  2,  1,  1,  1,...

30,  13,  7,  4,  4,  4,  3,  2,  1,  1,...

42,  16, 10,  5,  4,  4,  4,  3,  2,  1,...

56,  21, 12,  7,  4,  4,  4,  4,  3,  2,...

77,  28, 14, 10,  5,  4,  4,  4,  4,  3,...

101, 35, 16, 12,  7,  4,  4,  4,  4,  4,...

135, 43, 21, 13, 10,  5,  4,  4,  4,  4,...

176, 55, 27, 14, 12,  7,  4,  4,  4,  4,...

...

Column 1 is A000041 which starts: [1, 1], 2, 3, 5, 7, 11... The column contains only one plateau: [1, 1] which has level 1 and length 2.

Column 3 is A036820 which starts: [1, 1, 1, 1], 2, 3, [4, 4], 5, 7, 10... The column contains two plateaus: [1, 1, 1, 1], [4, 4], which have levels 1, 4 and lengths 4, 2.

Column 6 is A195850 which starts: [1, 1, 1, 1, 1, 1, 1], 2, 3, [4, 4, 4, 4, 4], 5, 7, 10, 12, [13, 13, 13], 14, 16, 21... The column contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13], which have levels 1, 4, 13 and lengths 7, 5, 3.

PROG

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

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

20 FOR k = 1 TO 10   'Column 1-10

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

110   FOR j = 1 TO 10

120     PRINT T(n, k);

130   NEXT k

140   PRINT

150 NEXT n

160 END

' Omar E. Pol, Jun 18 2012

CROSSREFS

Columns (1-10): A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.

For another version see A211970.

Cf. A057077, A195152, A210843.

Sequence in context: A109626 A182285 A160182 * A098824 A181651 A124032

Adjacent sequences:  A195822 A195823 A195824 * A195826 A195827 A195828

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, Sep 24 2011

STATUS

approved

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Last modified February 22 05:38 EST 2020. Contains 332115 sequences. (Running on oeis4.)