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EXAMPLE
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G.f.: A(x) = 1 - x^2 - 2*x^4 + 2*x^5 - 7*x^7 + 14*x^8 - 7*x^9 - 37*x^11 + 111*x^12 - 111*x^13 + 37*x^14 - 268*x^16 + 1072*x^17 - 1608*x^18 + 1072*x^19 - 268*x^20 - 2496*x^22 + 12480*x^23 - 24960*x^24 + 24960*x^25 - 12480*x^26 + 2496*x^27 - 28612*x^29 + 171672*x^30 + ...
The table of coefficients of x^k in A(x) / (1-x)^n, for n >= 0, begins:
[1, 0, -1, 0, -2, 2, 0, -7, 14, -7, 0, -37, 111,-111, 37, ...];
[1, 1, 0, 0, -2, 0, 0, -7, 7, 0, 0, -37, 74, -37, 0, ...];
[1, 2, 2, 2, 0, 0, 0, -7, 0, 0, 0, -37, 37, 0, 0, ...];
[1, 3, 5, 7, 7, 7, 7, 0, 0, 0, 0, -37, 0, 0, 0, ...];
[1, 4, 9, 16, 23, 30, 37, 37, 37, 37, 37, 0, 0, 0, 0, ...];
[1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268, 268, 268, 268, ...];
[1, 6, 20, 50, 103, 186, 306, 463, 657, 888, 1156,1424,1692,1960, 2228, ...];
[1, 7, 27, 77, 180, 366, 672, 1135, 1792, 2680, 3836,5260,6952,8912,11140, ...]; ...
illustrating the occurrence of zeros.
Note that the initial terms of the rows in the above table forms the rows of irregular triangle A127496.
TRIANGULAR FORM.
This sequence may be arranged into a triangle like so:
1,
0, -1,
0, -2, 2,
0, -7, 14, -7,
0, -37, 111, -111, 37,
0, -268, 1072, -1608, 1072, -268,
0, -2496, 12480, -24960, 24960, -12480, 2496,
0, -28612, 171672, -429180, 572240, -429180, 171672, -28612,
...
in which the g.f. of the rows equal -x * A107877(n) * (1-x)^(n-1) for n > 0.
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