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, 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

Offset: 1

Author

Carl McTague, Feb 03 2014

Keywords

• sign
• frac
• tabl

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.

Examples

			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,...
		

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
• Etienne Ghys and Andrew Ranicki, Signatures in algebra, topology, and dynamics, arXiv preprint arXiv:1512.09258 [math.AT], 2016. (Pg. 78). [From _Tom Copeland_, Aug 24 2019].
• Friedrich Hirzebruch and Matthias Kreck, The concept of genus in topology and complex analysis, Notices of the American Mathematical Society, 56 (6), p. 713-719, 2009, [From _Tom Copeland_, Aug 07 2018].
• Carl McTague, Computing Hirzebruch L-Polynomials.
• Jonas Stelzig, Differential forms and invariants of complex manifolds, arXiv:2503.05983 [math.AT], 2025. See p. 30.

Crossrefs

Cf. A171080.

Programs

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}]