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A237111
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Triangle read by rows: numerators of coefficients of the Hirzebruch L-polynomials L_n expressing the signature of a 4n-dimensional manifold in terms of its Pontrjagin numbers (as in Hirzebruch Signature Theorem).
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1
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1, 7, -1, 62, -13, 2, 381, -71, -19, 22, -3, 5110, -919, -336, 237, 127, -83, 10, 2828954, -503904, -159287, 122523, -40247, 86901, -33863, 8718, -27635, 12842, -1382, 3440220, -611266, -185150, 146256, -62274, 88137, -37290, 22027, 16696, -39341, 10692, -7978, 11880, -4322, 420
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OFFSET
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1,2
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COMMENTS
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The monomials of each polynomial L_n have been written in descending lexicographic order (where the exponent of p_1 is considered less significant than that of p_2, etc.) and over a common denominator. These denominators follow A171080.
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REFERENCES
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F. Hirzebruch, Topological methods in algebraic geometry. Third enlarged edition. Die Grundlehren der Mathematischen Wissenschaften, Band 131 Springer-Verlag New York, Inc., New York 1966, p. 12.
F. Hirzebruch, The signature theorem: reminiscences and recreation. Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), pp. 3-31. Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, N.J. 1971.
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LINKS
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EXAMPLE
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L_1 = p_1/3.
L_2 = (7 p_2-p_1^2)/45.
L_3 = (62 p_3-13 p_2 p_1+2 p_1^3)/945.
L_4 = (381 p_4-71 p_3 p_1-19 p_2^2+22 p_2 p_1^2-3 p_1^4)/14175.
L_5 = (5110 p_5-919 p_4 p_1-336 p_3 p_2+237 p_3 p_1^2+127 p_2^2 p_1-83 p_2 p_1^3+10 p_1^5)/467775.
The denominators 3,45,945,14175,467775 follow A171080.
The sequence of numerators thus begins 1,7,-1,62,-13,2,381,-71,-19,22,-3,5110,-919,-336,237,127,-83,10,...
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MATHEMATICA
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K[Q_, n_Integer] := Module[{z, x},
SymmetricReduction[
SeriesCoefficient[
Product[ComposeSeries[Series[Q[z], {z, 0, n}],
Series[Subscript[x, i]z, {z, 0, n}]], {i, 1, n}], n],
Table[Subscript[x, i], {i, 1, n}],
Table[Subscript[p, i], {i, 1, n}]][[1]] // FactorTerms];
Table[K[Sqrt[#]/Tanh[Sqrt[#]]&, n], {n, 1, 5}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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