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 A174739 Triangle read by rows, a partition number generator; A145006 * the diagonalized variant of A000041, (A174712). 2
 1, 1, 1, 0, 1, 2, 0, 0, 2, 3, -1, 0, 0, 3, 5, 0, -1, 0, 0, 5, 7, -1, 0, -2, 0, 0, 7, 11, 0, -1, 0, -3, 0, 0, 11, 15, 0, 0, -2, 0, -5, 0, 15, 22, 0, 0, 0, -3, 0, -7, 0, 0, 22, 30, 0, 0, 0, 0, -5, 0, -11, 0, 0, 30, 42, 1, 0, 0, 0, 0, -7, 0, -15, 0, 0, 42, 56, 0, 1, 0, 0, 0, 0, -11, 0, -22, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Row sums = the partition numbers, A000041 starting with offset 1. The triangle demonstrates an equivalency to Euler's pentagonal recurrence relation, such that sum of n-th row terms = rightmost term of next row, a partition number. Contribution from Gary W. Adamson, Mar 28 2010: (Start) A174739 is equivalent to Euler's pentagonal theorem in triangular form. For example, row 9 = (0, 0, -2, 0, -5, 0, 0, 15, 22) or: p(9) = 30 = p(8) + p(7) - p(4) - p(2). (End) LINKS Table of n, a(n) for n=1..90. FORMULA Given triangle A145006, delete the first "1", = triangle Q. With M = A174712, the diagonalize variant of the partition numbers, perform Q*M as infinite lower triangular matrices. EXAMPLE First few rows of the triangle = 1; 1, 1; 0, 1, 2; 0, 0, 2, 3; -1, 0, 0, 3, 5; 0, -1, 0, 0, 5, 7; -1, 0, -2, 0, 0, 7, 11; 0, -1, 0, -3, 0, 0, 11, 15; 0, 0, -2, 0, -5, 0, 0, 15, 22; 0, 0, 0, -3, 0, -7, 0, 0, 22, 30; 0, 0, 0, 0, -5, 0, -11, 0, 0, 30, 42; 1, 0, 0, 0, 0, -7, 0, -15, 0, 0, 42, 56; 0, 1, 0, 0, 0, 0, -11, 0, 22, 0, 0, 56, 77; 0, 0, 2, 0, 0, 0, 0, -15, 0, -30, 0, 0, 77, 101; 1, 0, 0, 3, 0, 0, 0, 0, -22, 0, -42, 0, 0, 101, 135; 0, 1, 0, 0, 5, 0, 0, 0, 0, -30, 0, -56, 0, 0, 135, 176; 0, 0, 2, 0, 0, 7, 0, 0, 0, 0, -42, 0, -77, 0, 0, 176, 231; 0, 0, 0, 3, 0, 0, 11, 0, 0, 0, 0, -56, 0, -101, 0, 0, 231, 297; ... CROSSREFS Cf. A000041, A145006, A174712 Sequence in context: A323886 A340829 A352551 * A280542 A340378 A274575 Adjacent sequences: A174736 A174737 A174738 * A174740 A174741 A174742 KEYWORD tabl,sign AUTHOR Gary W. Adamson, Mar 28 2010 STATUS approved

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Last modified June 20 13:44 EDT 2024. Contains 373527 sequences. (Running on oeis4.)