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A156890
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Triangle formed by coefficients of the expansion of p(x, n), where p(x,n) = (1+x-x^2)^(n+1)*Sum_{j >= 0} (j+1)^n*(-x + x^2)^j.
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5
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1, 1, 1, -1, 1, 1, -4, 5, -2, 1, 1, -11, 22, -23, 14, -3, 1, 1, -26, 92, -158, 145, -82, 32, -4, 1, 1, -57, 359, -906, 1265, -1135, 649, -238, 67, -5, 1, 1, -120, 1311, -4798, 9630, -12132, 10163, -5970, 2406, -620, 135, -6, 1, 1, -247, 4540, -24205, 66769, -113626, 131045, -106889, 62261, -26426, 8033, -1517, 268, -7, 1
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OFFSET
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0,7
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COMMENTS
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Row sums are equal to 1.
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LINKS
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G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened
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FORMULA
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T(n, k) = coefficients of the expansion of p(x, n), where p(x,n) = (1+x-x^2)^(n + 1)*Sum_{j >= 0} (j+1)^n*(-x + x^2)^j.
T(n, 1) = (-1)*A000295(n) for n >= 2. - G. C. Greubel, Jan 06 2022
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EXAMPLE
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Irregular triangle begins as:
1;
1;
1, -1, 1;
1, -4, 5, -2, 1;
1, -11, 22, -23, 14, -3, 1;
1, -26, 92, -158, 145, -82, 32, -4, 1;
1, -57, 359, -906, 1265, -1135, 649, -238, 67, -5, 1;
1, -120, 1311, -4798, 9630, -12132, 10163, -5970, 2406, -620, 135, -6, 1;
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MATHEMATICA
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p[x_, n_]:= ((1+x-x^2)^(n+1))*Sum[(j+1)^n*(-x+x^2)^j, {j, 0, Infinity}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten
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PROG
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(Sage)
def T(n, k): return ( (1+x-x^2)^(n+1)*sum((j+1)^n*(x^2-x)^j for j in (0..2*n+1)) ).series(x, 2*n+3).list()[k]
[1]+flatten([[T(n, k) for k in (0..2*n-2)] for n in (0..12)]) # G. C. Greubel, Jan 06 2022
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CROSSREFS
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Cf. A000295, A156896, A156901, A156918.
Sequence in context: A157784 A274615 A258895 * A320480 A163531 A336199
Adjacent sequences: A156887 A156888 A156889 * A156891 A156892 A156893
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KEYWORD
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tabf,sign
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AUTHOR
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Roger L. Bagula, Feb 17 2009
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EXTENSIONS
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Edited by G. C. Greubel, Jan 06 2022
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STATUS
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approved
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