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A175003 Triangle read by rows demonstrating Euler's pentagonal theorem for partition numbers.

%I A175003 #40 Apr 02 2017 14:50:01
%S A175003 1,1,1,2,1,3,2,5,3,-1,7,5,-1,11,7,-2,-1,15,11,-3,-1,22,15,-5,-2,30,22,
%T A175003 -7,-3,42,30,-11,-5,56,42,-15,-7,1,77,56,-22,-11,1,101,77,-30,-15,2,
%U A175003 135,101,-42,-22,3,1,176,135,-56,-30,5,1,231,176,-77,-42,7,2
%N A175003 Triangle read by rows demonstrating Euler's pentagonal theorem for partition numbers.
%C A175003 Row sums = A000041 starting with offset 1.
%C A175003 Sum of n-th row terms = leftmost term of next row, such that terms in each row demonstrate Euler's pentagonal theorem.
%C A175003 Let Q = triangle A027293 with partition numbers in each column.
%C A175003 Let M = a diagonalized variant of A080995 as the characteristic function of the generalized pentagonal numbers starting with offset 1: (1, 1, 0, 0, 1,...)
%C A175003 Sign the 1's: (++--++...) getting (1, 1, 0, 0, -1, 0, -1,...) which is the diagonal of matrix M, (as an infinite lower triangular matrix with the rest zeros).
%C A175003 Triangle A175003 = Q*M, with deleted zeros.
%C A175003 Column k starts at row A001318(k). - _Omar E. Pol_, Sep 21 2011
%C A175003 From _Omar E. Pol_, Apr 22 2014: (Start)
%C A175003 Row n has length A235963(n).
%C A175003 For Euler's pentagonal theorem for the sum of divisors see A238442.
%C A175003 Note that both of Euler's pentagonal theorems refer to generalized pentagonal numbers (A001318), not to pentagonal numbers (A000326). (End)
%F A175003 T(n,k) = A057077(k-1)*A000041(A195310(n,k)), n >= 1, k >= 1. - _Omar E. Pol_, Sep 21 2011
%e A175003 Triangle begins:
%e A175003     1;
%e A175003     1,   1;
%e A175003     2,   1;
%e A175003     3,   2;
%e A175003     5,   3,  -1;
%e A175003     7,   5,  -1;
%e A175003    11,   7,  -2,  -1;
%e A175003    15,  11,  -3,  -1;
%e A175003    22,  15,  -5,  -2;
%e A175003    30,  22,  -7,  -3;
%e A175003    42,  30, -11,  -5;
%e A175003    56,  42, -15,  -7,   1;
%e A175003    77,  56, -22, -11,   1;
%e A175003   101,  77, -30, -15,   2;
%e A175003   ...
%Y A175003 Cf. A000041, A080995, A027293, A238442.
%K A175003 tabf,sign
%O A175003 1,4
%A A175003 _Gary W. Adamson_, Apr 03 2010
%E A175003 Corrected and extended by _Omar E. Pol_, Feb 14 2013