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A156985
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Triangle formed by coefficients of the expansion of p(x,n) = (1-x)^(2*n + 1)*Sum_{j >= 0} (1 +j +j^2)^n * x^j.
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1
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1, 1, 0, 1, 1, 4, 14, 4, 1, 1, 20, 175, 328, 175, 20, 1, 1, 72, 1708, 9784, 17190, 9784, 1708, 72, 1, 1, 232, 14189, 199616, 884498, 1431728, 884498, 199616, 14189, 232, 1, 1, 716, 108250, 3353948, 31986447, 115907544, 176287788, 115907544, 31986447, 3353948, 108250, 716, 1
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OFFSET
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0,6
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LINKS
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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)^(2*n + 1)*Sum_{j >= 0} (1 +j +j^2)^n * x^j.
Sum_{k=0..2*n} T(n, k) = A010050(n).
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EXAMPLE
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Irregular triangle begins as:
1;
1, 0, 1;
1, 4, 14, 4, 1;
1, 20, 175, 328, 175, 20, 1;
1, 72, 1708, 9784, 17190, 9784, 1708, 72, 1;
1, 232, 14189, 199616, 884498, 1431728, 884498, 199616, 14189, 232, 1;
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MATHEMATICA
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p[x_, n_] = (1-x)^(2*n+1)*Sum[(1+k+k^2)^n*x^k, {k, 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)^(2*n+1)*sum((j^2+j+1)^n*x^j for j in (0..2*n+1)) ).series(x, 2*n+2).list()[k]
flatten([1]+[[T(n, k) for k in (0..2*n)] for n in (1..12)]) # G. C. Greubel, Jan 07 2022
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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