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 A175003 Triangle read by rows demonstrating Euler's pentagonal theorem for partition numbers. 26
 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, -7, -3, 42, 30, -11, -5, 56, 42, -15, -7, 1, 77, 56, -22, -11, 1, 101, 77, -30, -15, 2, 135, 101, -42, -22, 3, 1, 176, 135, -56, -30, 5, 1, 231, 176, -77, -42, 7, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Row sums = A000041 starting with offset 1. Sum of n-th row terms = leftmost term of next row, such that terms in each row demonstrate Euler's pentagonal theorem. Let Q = triangle A027293 with partition numbers in each column. 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,...) 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). Triangle A175003 = Q*M, with deleted zeros. Column k starts at row A001318(k). - Omar E. Pol, Sep 21 2011 From Omar E. Pol, Apr 22 2014: (Start) Row n has length A235963(n). For Euler's pentagonal theorem for the sum of divisors see A238442. Note that both of Euler's pentagonal theorems refer to generalized pentagonal numbers (A001318), not to pentagonal numbers (A000326). (End) LINKS FORMULA T(n,k) = A057077(k-1)*A000041(A195310(n,k)), n >= 1, k >= 1. - Omar E. Pol, Sep 21 2011 EXAMPLE Triangle begins:     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,  -7,  -3;    42,  30, -11,  -5;    56,  42, -15,  -7,   1;    77,  56, -22, -11,   1;   101,  77, -30, -15,   2;   ... CROSSREFS Cf. A000041, A080995, A027293, A238442. Sequence in context: A240867 A242363 A050360 * A176431 A045747 A308984 Adjacent sequences:  A175000 A175001 A175002 * A175004 A175005 A175006 KEYWORD tabf,sign AUTHOR Gary W. Adamson, Apr 03 2010 EXTENSIONS Corrected and extended by Omar E. Pol, Feb 14 2013 STATUS approved

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Last modified February 24 10:27 EST 2020. Contains 332209 sequences. (Running on oeis4.)