This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A211970 #43 Feb 16 2025 08:33:17 %S A211970 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 A211970 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 A211970 1,1,1,112,42,13,5,4,2,1,1,1,1,1,1 %N A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem. %C A211970 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 A211970 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 A211970 ======================================================== %C A211970 . Column k of %C A211970 . this square %C A211970 . Generalized Triangle Triangle array A211970 %C A211970 k m m-gonal "A" "B" [row sums of %C A211970 . numbers triangle "B" %C A211970 . (if k>=1) with a(0)=1, %C A211970 . if k >= 0] %C A211970 ======================================================== %C A211970 0 4 A008794 - - A211971 %C A211970 1 5 A001318 A195310 A175003 A000041 %C A211970 2 6 A000217 A195826 A195836 A006950 %C A211970 3 7 A085787 A195827 A195837 A036820 %C A211970 4 8 A001082 A195828 A195838 A195848 %C A211970 5 9 A118277 A195829 A195839 A195849 %C A211970 6 10 A074377 A195830 A195840 A195850 %C A211970 7 11 A195160 A195831 A195841 A195851 %C A211970 8 12 A195162 A195832 A195842 A195852 %C A211970 9 13 A195313 A195833 A195843 A196933 %C A211970 10 14 A195818 A210944 A210954 A210964 %C A211970 ... %C A211970 It appears that column 2 of the square array is A006950. %C A211970 It appears that column 3 of the square array is A036820. %C A211970 The partial sums of column 0 give A015128. - _Omar E. Pol_, Feb 09 2014 %H A211970 L. Euler, <a href="http://www.math.dartmouth.edu/~euler/docs/originals/E542.pdf">De mirabilibus proprietatibus numerorum pentagonalium</a> %H A211970 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 A211970 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a> %F A211970 T(n,k) = A211971(n), if k = 0. %F A211970 T(n,k) = A195825(n,k), if k >= 1. %e A211970 Array begins: %e A211970 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... %e A211970 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... %e A211970 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... %e A211970 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, ... %e A211970 6, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, ... %e A211970 10, 7, 4, 3, 2, 1, 1, 1, 1, 1, 1, ... %e A211970 16, 11, 5, 4, 3, 2, 1, 1, 1, 1, 1, ... %e A211970 24, 15, 7, 4, 4, 3, 2, 1, 1, 1, 1, ... %e A211970 36, 22, 10, 5, 4, 4, 3, 2, 1, 1, 1, ... %e A211970 54, 30, 13, 7, 4, 4, 4, 3, 2, 1, 1, ... %e A211970 78, 42, 16, 10, 5, 4, 4, 4, 3, 2, 1, ... %e A211970 112, 56, 21, 12, 7, 4, 4, 4, 4, 3, 2, ... %e A211970 160, 77, 28, 14, 10, 5, 4, 4, 4, 4, 3, ... %e A211970 224, 101, 35, 16, 12, 7, 4, 4, 4, 4, 4, ... %e A211970 312, 135, 43, 21, 13, 10, 5, 4, 4, 4, 4, ... %e A211970 432, 176, 55, 27, 14, 12, 7, 4, 4, 4, 4, ... %e A211970 ... %o A211970 (GW-BASIC)' A program (with two A-numbers) for the square array of the example section. %o A211970 10 DIM A057077(100), A195152(15,10), T(15,10) %o A211970 20 FOR K = 0 TO 10 'Column 0-10 %o A211970 30 T(0,K) = 1 'Row 0 %o A211970 40 FOR N = 1 TO 15 'Rows 1-15 %o A211970 50 FOR J = 1 TO N %o A211970 60 IF A195152(J,K) <= N THEN T(N,K) = T(N,K) + A057077(J-1) * T(N - A195152(J,K),K) %o A211970 70 NEXT J %o A211970 80 NEXT N %o A211970 90 NEXT K %o A211970 100 FOR N = 0 TO 15: FOR K = 0 TO 10 %o A211970 110 PRINT T(N,K); %o A211970 120 NEXT K: PRINT: NEXT N %o A211970 130 END %Y A211970 Columns (0-10): A211971, A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964. %Y A211970 For another version see A195825. %Y A211970 Cf. A057077, A195152. %K A211970 nonn,tabl %O A211970 0,4 %A A211970 _Omar E. Pol_, Jun 10 2012