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 A341418 #33 Mar 22 2025 02:12:56
%S A341418 1,1,1,0,2,1,0,1,3,1,-1,0,3,4,1,0,-2,1,6,5,1,-1,-2,-3,4,10,6,1,0,-2,
%T A341418 -6,-3,10,15,7,1,0,-2,-6,-12,0,20,21,8,1,0,1,-6,-16,-19,9,35,28,9,1,0,
%U A341418 0,0,-16,-35,-24,28,56,36,10,1,1,2,3,-6,-40,-65,-21,62,84,45,11,1
%N A341418 Triangle read by rows: T(n, m) gives the sum of the weights of weighted compositions of n with m parts from generalized pentagonal numbers {A001318(k)}_{k>=1}.
%C A341418 The sums of row n are given in A000041(n), for n >= 1 (number of partitions).
%C A341418 A differently signed triangle is A047265.
%C A341418 One could add a column m = 0 starting at n = 0 with T(0, 0) = 1 and T(n, 0) = 0 otherwise, by including the empty partition with no parts.
%C A341418 For the weights w of positive integer numbers n see a comment in A339885. It is w(n) = -A010815(n), for n >= 0. Also w(n) = A257628(n), for n >= 1.
%C A341418 The weight of a composition is the one of the respective partition, obtained by the product of the weights of the parts.
%C A341418 That the row sums give the number of partitions follows from the pentagonal number theorem. See also the Apr 04 2013 conjecture in A000041 by _Gary W. Adamson_, and the hint for the proof by _Joerg Arndt_. The INVERT map of A = {1, 1, 0, 0, -5, -7, ...}, with offset 1, gives the A000041(n) numbers, for n >= 0.
%C A341418 If the above mentioned column for m = 0, starting at n = 0 is added this is an ordinary convolution triangle of the Riordan type R(1, f(x)), with f(x) = -(Product_{j>=1} (1 - x^j) - 1), generating {A257628(n)}_{n>=0}. See the formulae below. - _Wolfdieter Lang_, Feb 16 2021
%H A341418 Wikipedia, <a href="https://www.wikipedia.org/wiki/Pentagonal_number_theorem">Pentagonal number theorem</a>.
%F A341418 T(n, m) = Sum_{j=1..p(n,m)} w(Part(n, m, j))*M0(n, m, j), where p(n, m) = A008284(n, m), M0(n, m, j) are the multinomials from A048996, i.e., m!/Prod_{k=1..m} e(n,m,j,k)! with the exponents of the parts, and the ternary weight of the j-th partition of n with m parts Part(n,m,j), in Abramowitz-Stegun order, is defined as the product of the weights of the parts, using w(n) = -A010815(n), for n >= 1, and m = 1, 2, ..., n.
%F A341418 From _Wolfdieter Lang_, Feb 16 2021: (Start)
%F A341418 G.f. column m: G(m, x) = ( -(Product_{j>=1} (1 - x^j) - 1) )^m, for m >= 1.
%F A341418 G.f. of row polynomials R(n, x) = Sum_{m=1..n}, that is g. f. of the triangle:
%F A341418 GfT(z, x) = 1/(1 - x*G(1, z)) - 1. Riordan triangle (without m = 0 column). (End)
%e A341418 The triangle T(n, m) begins:
%e A341418 n\m 1 2 3 4 5 6 7 8 9 10 11 12 ... A000041
%e A341418 --------------------------------------------------------
%e A341418 1: 1 1
%e A341418 2: 1 1 2
%e A341418 3: 0 2 1 3
%e A341418 4: 0 1 3 1 5
%e A341418 5: -1 0 3 4 1 7
%e A341418 6: 0 -2 1 6 5 1 11
%e A341418 7: -1 -2 -3 4 10 6 1 15
%e A341418 8: 0 -2 -6 -3 10 15 7 1 22
%e A341418 9: 0 -2 -6 -12 0 20 21 8 1 30
%e A341418 10: 0 1 -6 -16 -19 9 35 28 9 1 42
%e A341418 11: 0 0 0 -16 -35 -24 28 56 36 10 1 56
%e A341418 12: 1 2 3 -6 -40 -65 -21 62 84 45 11 1 77
%e A341418 ...
%e A341418 For instance the case n = 6: The relevant weighted partitions with parts from the pentagonal numbers and number of compositions are: m = 2: 2*(1,-5) = -2*(1,5), m = 3: 1*(2^3), m = 4: 3*(1^2,2^2), m = 5: 1*(1^4,2), m = 6: 1*(1^6). The other partitions have weight 0.
%p A341418 # Using function PMatrix from A357368. Adds a row and a column for n, m = 0.
%p A341418 PMatrix(14, proc(n) 24*n+1; if issqr(%) then sqrt(%); -(-1)^irem(iquo(%+irem(%,6),6),2) else 0 fi end); # _Peter Luschny_, Oct 06 2022
%t A341418 nmax = 12;
%t A341418 col[m_] := col[m] = (-(Product[(1-x^j), {j, 1, nmax}]-1))^m // CoefficientList[#, x]&;
%t A341418 T[n_, m_] := col[m][[n+1]];
%t A341418 Table[T[n, m], {n, 1, nmax}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Oct 23 2023 *)
%Y A341418 Cf. A000041, A008284, A010815, A047265, A257628, -A307059 (alternating row sums), A339885 (for partitions).
%K A341418 sign,tabl,easy
%O A341418 1,5
%A A341418 _Wolfdieter Lang_, Feb 15 2021