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A237111 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). 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

REFERENCES

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.

LINKS

Carl McTague, Table of n, a(n) for n = 1..372

Carl McTague, Computing Hirzebruch L-Polynomials.

EXAMPLE

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,...

MATHEMATICA

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}]

CROSSREFS

Sequence in context: A144450 A051339 A134141 * A221367 A110788 A100254

Adjacent sequences:  A237108 A237109 A237110 * A237112 A237113 A237114

KEYWORD

sign,frac,tabl

AUTHOR

Carl McTague, Feb 03 2014

STATUS

approved

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Last modified December 21 04:58 EST 2014. Contains 252293 sequences.