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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).

%I A237111 #50 Mar 13 2025 12:14:15
%S A237111 1,7,-1,62,-13,2,381,-71,-19,22,-3,5110,-919,-336,237,127,-83,10,
%T A237111 2828954,-503904,-159287,122523,-40247,86901,-33863,8718,-27635,12842,
%U A237111 -1382,3440220,-611266,-185150,146256,-62274,88137,-37290,22027,16696,-39341,10692,-7978,11880,-4322,420
%N 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).
%C A237111 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.
%D A237111 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.
%D A237111 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.
%H A237111 Carl McTague, <a href="/A237111/b237111.txt">Table of n, a(n) for n = 1..372</a>
%H A237111 Etienne Ghys and Andrew Ranicki, <a href="https://arxiv.org/abs/1512.09258">Signatures in algebra, topology, and dynamics</a>, arXiv preprint arXiv:1512.09258 [math.AT], 2016. (Pg. 78). [From _Tom Copeland_, Aug 24 2019].
%H A237111 Friedrich Hirzebruch and Matthias Kreck, <a href="https://www.ams.org/notices/200906/rtx090600713p.pdf">The concept of genus in topology and complex analysis</a>, Notices of the American Mathematical Society, 56 (6), p. 713-719, 2009, [From _Tom Copeland_, Aug 07 2018].
%H A237111 Carl McTague, <a href="http://www.mctague.org/carl/blog/2014/01/05/computing-L-polynomials/">Computing Hirzebruch L-Polynomials</a>.
%H A237111 Jonas Stelzig, <a href="https://arxiv.org/abs/2503.05983">Differential forms and invariants of complex manifolds</a>, arXiv:2503.05983 [math.AT], 2025. See p. 30.
%e A237111 L_1 = p_1/3.
%e A237111 L_2 = (7 p_2-p_1^2)/45.
%e A237111 L_3 = (62 p_3-13 p_2 p_1+2 p_1^3)/945.
%e A237111 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.
%e A237111 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.
%e A237111 The denominators 3,45,945,14175,467775 follow A171080.
%e A237111 The sequence of numerators thus begins 1,7,-1,62,-13,2,381,-71,-19,22,-3,5110,-919,-336,237,127,-83,10,...
%t A237111 K[Q_,n_Integer] := Module[{z,x},
%t A237111    SymmetricReduction[
%t A237111       SeriesCoefficient[
%t A237111        Product[ComposeSeries[Series[Q[z],{z,0,n}],
%t A237111          Series[Subscript[x,i]z,{z,0,n}]],{i,1,n}],n],
%t A237111       Table[Subscript[x,i],{i,1,n}],
%t A237111       Table[Subscript[p,i],{i,1,n}]][[1]] // FactorTerms];
%t A237111 Table[K[Sqrt[#]/Tanh[Sqrt[#]]&, n],{n,1,5}]
%Y A237111 Cf. A171080.
%K A237111 sign,frac,tabl
%O A237111 1,2
%A A237111 _Carl McTague_, Feb 03 2014