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A175010 Triangle generated from INVERT transforms of variants of A080995.

%I A175010 #12 Nov 12 2017 03:58:23
%S A175010 1,1,1,1,1,1,1,1,1,2,1,1,1,1,3,1,1,1,1,2,5,1,1,1,1,1,4,6,1,1,1,1,1,2,
%T A175010 6,9,1,1,1,1,1,1,4,8,12,1,1,1,1,1,1,2,6,12,16,1,1,1,1,1,1,1,4,8,19,18,
%U A175010 1,1,1,1,1,1,1,2,6,11,28,23
%N A175010 Triangle generated from INVERT transforms of variants of A080995.
%C A175010 Row sums = A000041 starting with offset 1: (1, 1, 2, 3, 5, 7, 11, 15, ...).
%C A175010 The INVERTi transform of A000041 starting with offset 1 follows from the definition of the INVERT transform, given 1/p(x) = A010815.
%F A175010 Given the INVERTi transform of the partition numbers starting with offset 1 = a signed variant of A080995 such that Q = (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, ...).
%F A175010 Construct an array in which k-th row (k=1,2,3,...) = the INVERT transform of Q(x^k), i.e., where polcoeff Q(x) is interleaved with 0,1,2,3,... zeros.
%F A175010 Take finite differences of the array terms starting with the last "1" going from the bottom to top, becoming rows of triangle A175010.
%e A175010 First few rows of the array:
%e A175010 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101,
%e A175010 1, 1, 2, 3, 4,  6,  9, 13, 18, 26, 38, 54,  76,
%e A175010 1, 1, 1, 2, 3,  4,  5,  7, 10, 14, 19, 26,  35,
%e A175010 1, 1, 1, 1, 2,  3,  4,  5,  6,  8, 11, 15,  20,
%e A175010 1, 1, 1, 1, 1,  2,  3,  4,  5,  6,  7,  9,  12,
%e A175010 1, 1, 1, 1, 1,  1,  2,  3,  4,  5,  6,  7,   8,
%e A175010 ...
%e A175010 Taking finite differences from the bottom starting with the top "1", we obtain rows of the triangle:
%e A175010 1;
%e A175010 1, 1;
%e A175010 1, 1, 1;
%e A175010 1, 1, 1, 2;
%e A175010 1, 1, 1, 1, 3;
%e A175010 1, 1, 1, 1, 2, 5;
%e A175010 1, 1, 1, 1, 1, 4, 6;
%e A175010 1, 1, 1, 1, 1, 2, 6, 9;
%e A175010 1, 1, 1, 1, 1, 1, 4, 8, 12;
%e A175010 1, 1, 1, 1, 1, 1, 2, 6, 12, 16;
%e A175010 1, 1, 1, 1, 1, 1, 1, 4,  8, 19, 18;
%e A175010 1, 1, 1, 1, 1, 1, 1, 2,  6, 11, 28, 23;
%e A175010 1, 1, 1, 1, 1, 1, 1, 1,  4,  8, 15, 41, 25;
%e A175010 1, 1, 1, 1, 1, 1, 1, 1,  2,  6, 10, 22, 61, 26;
%e A175010 ...
%e A175010 Example: Row 2 = INVERT transform of Q(x^2), (i.e., Q(x) interleaved with one zero between terms).
%Y A175010 Cf. A000041, A080995, A010815.
%K A175010 nonn,tabl
%O A175010 1,10
%A A175010 _Gary W. Adamson_, Apr 03 2010