A351400 Decimal expansion of e * erf(1), where erf is the error function.

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

Offset: 1

Author

Amiram Eldar, Feb 10 2022

Keywords

• nonn
• cons

Comments

The sum of reciprocals of the factorials of the positive half-integers.

Examples

			2.29069825230323823094953712686214731693708759053570...
		

References

• Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, and Sergei Rogosin, Mittag-Leffler Functions, Related Topics and Applications, New York, NY: Springer, 2020. See p. 94, eq. (4.12.9.5).
• Constantin Milici, Gheorghe Drăgănescu, and J. Tenreiro Machado, Fractional Differential Equations, Introduction to Fractional Differential Equations, Springer, Cham, 2019. See p. 12, eq. (1.9).

Links

• Eric Weisstein's World of Mathematics, Erf.
• Eric Weisstein's World of Mathematics, Mittag-Leffler Function.
• Wikipedia, Mittag-Leffler function.

Crossrefs

Cf. A000079, A001113, A001147, A087197, A099286, A125961, A351401.

Programs

Maple

evalf(exp(1)*erf(1), 120);  # Alois P. Heinz, Feb 10 2022

Mathematica

RealDigits[E * Erf[1], 10, 100][[1]]

PARI

exp(1)*(1 - erfc(1)) \\ Michel Marcus, Feb 10 2022

Formula

Equals Sum_{k>=0} 1/(k + 1/2)! = Sum_{k>=1} 1/Gamma(k + 1/2).

Equals E_{1, 3/2}(1), where E_{a,b}(z) is the two-parameter Mittag-Leffler function.

Equals (1/sqrt(Pi)) * Sum_{k>=1} 2^k/(2*k-1)!! = (1/sqrt(Pi)) * Sum_{k>=1} A000079(k)/A001147(k).

Equals A001113 * A099286.

Equals A087197 * A125961.