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%I #98 May 17 2024 15:24:10
%S 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,
%T 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,
%U 1,56,16,7,4,3,1,1,1,1,1,1,1,77,21,10,4
%N Square array T(n,k) read by antidiagonals, n>=0, k>=1, which arises from a generalization of Euler's Pentagonal Number Theorem.
%C 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).
%C 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):
%C ========================================================
%C . Column k of
%C . this square
%C . Generalized Triangle Triangle array A195825
%C k m m-gonal "A" "B" [row sums of
%C . numbers triangle "B"
%C . with a(0)=1]
%C ========================================================
%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 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
%H Leonhard Euler, <a href="http://eulerarchive.maa.org/pages/E542.html">De mirabilibus proprietatibus numerorum pentagonalium</a>
%H Leonhard 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>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pentagonal_number_theorem">Pentagonal number theorem</a>
%F Column k is asymptotic to exp(Pi*sqrt(2*n/(k+2))) / (8*sin(Pi/(k+2))*n). - _Vaclav Kotesovec_, Aug 14 2017
%e Array begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, ...
%e 7, 4, 3, 2, 1, 1, 1, 1, 1, 1, ...
%e 11, 5, 4, 3, 2, 1, 1, 1, 1, 1, ...
%e 15, 7, 4, 4, 3, 2, 1, 1, 1, 1, ...
%e 22, 10, 5, 4, 4, 3, 2, 1, 1, 1, ...
%e 30, 13, 7, 4, 4, 4, 3, 2, 1, 1, ...
%e 42, 16, 10, 5, 4, 4, 4, 3, 2, 1, ...
%e 56, 21, 12, 7, 4, 4, 4, 4, 3, 2, ...
%e 77, 28, 14, 10, 5, 4, 4, 4, 4, 3, ...
%e 101, 35, 16, 12, 7, 4, 4, 4, 4, 4, ...
%e 135, 43, 21, 13, 10, 5, 4, 4, 4, 4, ...
%e 176, 55, 27, 14, 12, 7, 4, 4, 4, 4, ...
%e ...
%e 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.
%e 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.
%e 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.
%o (GW-BASIC)' A program (with two A-numbers) for the table of example section.
%o 10 DIM A057077(100), A195152(15,10), T(15,10)
%o 20 FOR k = 1 TO 10 'Column 1-10
%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
%o 110 FOR j = 1 TO 10
%o 120 PRINT T(n,k);
%o 130 NEXT k
%o 140 PRINT
%o 150 NEXT n
%o 160 END
%o 170 '_Omar E. Pol_, Jun 18 2012
%Y Columns (1-10): A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
%Y For another version see A211970.
%Y Cf. A057077, A195152, A210843.
%K nonn,tabl
%O 0,4
%A _Omar E. Pol_, Sep 24 2011
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