A261745 Decimal expansion of -sm(-1), where sm(t) is the Dixonian elliptic function sm(t).

1, 2, 0, 5, 4, 1, 5, 1, 5, 1, 4, 0, 2, 9, 8, 3, 1, 5, 4, 8, 3, 1, 4, 1, 1, 3, 7, 5, 7, 8, 4, 4, 8, 8, 0, 1, 2, 0, 7, 2, 7, 0, 4, 1, 9, 1, 8, 8, 2, 2, 4, 9, 5, 8, 1, 0, 9, 3, 2, 7, 1, 8, 2, 3, 5, 4, 4, 7, 6, 4, 8, 8, 1, 0, 6, 5, 5, 1, 1, 2, 5, 5, 6, 3, 2, 1, 7, 0, 3, 6, 5, 2, 2, 9, 3, 7, 9, 8, 9, 0, 8, 0, 6, 7, 9

Offset: 1

Author

Jean-François Alcover, Aug 30 2015

Keywords

• nonn
• cons
• easy

Comments

In the context of particle physics and in the case of a Yule branching process with two types of particles, this constant appears in the asymptotic expression of the probability that all particles be of the second type at time t as exp(-t)*smh(1).

Examples

			1.205415151402983154831411375784488012072704191882249581...
		

Links

• Eric van Fossen Conrad and Philippe Flajolet, The Fermat cubic, elliptic functions, continued fractions and a combinatorial excursion, Séminaire Lotharingien de Combinatoire 54 (2006), Article B54g, page 20.
• Eric Weisstein's MathWorld, Weierstrass Elliptic Function

Crossrefs

Cf. A104133, A104134.

Programs

Mathematica

sm[z_] := 6*WeierstrassP[z, {0, 1/27}]/(1 - 3*WeierstrassPPrime[z, {0, 1/27}]); N[-sm[-1], 105] // RealDigits // First
(* or, without using the Weierstrass P function: *) nint[y_?NumericQ] := NIntegrate[1/(1 + w^3)^(2/3), {w, 0, y}, WorkingPrecision -> 105]; smh[t_] := y /. FindRoot[nint[y] == t, {y, t}, WorkingPrecision -> 105]; N[smh[1], 105] // RealDigits // First