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%I #39 Apr 10 2020 14:33:02
%S 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,
%T 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,
%U 1,1,1,112,42,13,5,4,2,1,1,1,1,1,1
%N Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.
%C 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...
%C 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):
%C ========================================================
%C . Column k of
%C . this square
%C . Generalized Triangle Triangle array A211970
%C k m m-gonal "A" "B" [row sums of
%C . numbers triangle "B"
%C . (if k>=1) with a(0)=1,
%C . if k >= 0]
%C ========================================================
%C 0 4 A008794 - - A211971
%C 1 5 A001318 A195310 A175003 A000041
%C 2 6 A000217 A195826 A195836 A006950
%C 3 7 A085787 A195827 A195837 A036820
%C 4 8 A001082 A195828 A195838 A195848
%C 5 9 A118277 A195829 A195839 A195849
%C 6 10 A074377 A195830 A195840 A195850
%C 7 11 A195160 A195831 A195841 A195851
%C 8 12 A195162 A195832 A195842 A195852
%C 9 13 A195313 A195833 A195843 A196933
%C 10 14 A195818 A210944 A210954 A210964
%C ...
%C It appears that column 2 of the square array is A006950.
%C It appears that column 3 of the square array is A036820.
%C The partial sums of column 0 give A015128. - _Omar E. Pol_, Feb 09 2014
%H L. Euler, <a href="http://www.math.dartmouth.edu/~euler/docs/originals/E542.pdf">De mirabilibus proprietatibus numerorum pentagonalium</a>
%H L. Euler, <a href="https://arxiv.org/abs/math/0505373">On the remarkable properties of the pentagonal numbers</a>, arXiv:math/0505373 [math.HO], 2005.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>
%F T(n,k) = A211971(n), if k = 0.
%F T(n,k) = A195825(n,k), if k >= 1.
%e Array begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 6, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, ...
%e 10, 7, 4, 3, 2, 1, 1, 1, 1, 1, 1, ...
%e 16, 11, 5, 4, 3, 2, 1, 1, 1, 1, 1, ...
%e 24, 15, 7, 4, 4, 3, 2, 1, 1, 1, 1, ...
%e 36, 22, 10, 5, 4, 4, 3, 2, 1, 1, 1, ...
%e 54, 30, 13, 7, 4, 4, 4, 3, 2, 1, 1, ...
%e 78, 42, 16, 10, 5, 4, 4, 4, 3, 2, 1, ...
%e 112, 56, 21, 12, 7, 4, 4, 4, 4, 3, 2, ...
%e 160, 77, 28, 14, 10, 5, 4, 4, 4, 4, 3, ...
%e 224, 101, 35, 16, 12, 7, 4, 4, 4, 4, 4, ...
%e 312, 135, 43, 21, 13, 10, 5, 4, 4, 4, 4, ...
%e 432, 176, 55, 27, 14, 12, 7, 4, 4, 4, 4, ...
%e ...
%o (GWbasic)' A program (with two A-numbers) for the square array of the example section.
%o 10 DIM A057077(100), A195152(15,10), T(15,10)
%o 20 FOR K = 0 TO 10 'Column 0-10
%o 30 T(0,K) = 1 'Row 0
%o 40 FOR N = 1 TO 15 'Rows 1-15
%o 50 FOR J = 1 TO N
%o 60 IF A195152(J,K) <= N THEN T(N,K) = T(N,K) + A057077(J-1) * T(N - A195152(J,K),K)
%o 70 NEXT J
%o 80 NEXT N
%o 90 NEXT K
%o 100 FOR N = 0 TO 15: FOR K = 0 TO 10
%o 110 PRINT T(N,K);
%o 120 NEXT K: PRINT: NEXT N
%o 130 END
%Y Columns (0-10): A211971, A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
%Y For another version see A195825.
%Y Cf. A057077, A195152.
%K nonn,tabl
%O 0,4
%A _Omar E. Pol_, Jun 10 2012
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