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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
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]
========================================================
...
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
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
(GW-BASIC)' 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
For another version see A195825.
Sequence in context: A367559 A099020 A179438 * A089688 A229706 A319421
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Jun 10 2012
STATUS
approved