2,3

The array contains the coefficients for a normalized Schwarzian of univalent functions: S(f(x)) = (x^2/6) { D^2 ln(f(x)) - (1/2) [D ln(f(x))]^2 } with f(x) = 1 / [1 - c(.) x]^2 = 1 + 2 c(1) x + 3 c(2) x^2 + ....

S(f(x)) = P(2,c) x^2 + P(3,c) x^3 + P(4,c) x^4 + ..., where the multinomials P(n,c) are the Neretin multinomials with an additional factor of 2. Conjecture: the P(n,c) contain only integer coefficients.

R. Hidalgo, I. Markina, A. Vasil'ev, Finite dimensional grading of the Virasoro algebra, Georg. Math. J. 14 (2007), 419-434.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972

R. Hidalgo, I. Markina and A. Vasil'ev, Finite dimensional grading of the Virasoro algebra

A. Kirillov, Geometric approach to discrete series of unireps for Vir.

See references for recurrences and lowering operators.

.. P(0,c) = 0

.. P(1,c) = 0

.. P(2,c) = c(2) - c(1)^2

.. P(3,c) = 4 c(3) - 8 c(2)c(1) + 4 c(1)^3 = 4 3' - 8 2'1' + 4 1'^3

.. P(4,c) = 10 4' - 20 3'1' - 12 2'^2 + 34 2'1'^2 - 12 1'^4

.. P(5,c) = 20 5' - 40 1'4' - 52 2'3' + 72 3'1'^2 + 84 2'^2 1'- 116 2'1'^3 + 32 1'^5

The partitions are arranged in the order of those of Abramowitz and Stegun on pg. 831.

Sequence in context: A165267 A092159 A141402 this_sequence A010298 A155719 A059159

Adjacent sequences: A145897 A145898 A145899 this_sequence A145901 A145902 A145903

easy,sign,tabf

Tom Copeland (tcjpn(AT)msn.com), Oct 22 2008

Corrected typo in first author's name in a link and added reference to the journal article. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 04 2008