A351400 - OEIS

%I #16 Dec 25 2024 04:05:02

%S 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,

%T 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,

%U 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

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

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

%D 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).

%D 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).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Erf.html">Erf</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Mittag-LefflerFunction.html">Mittag-Leffler Function</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Mittag-Leffler_function">Mittag-Leffler function</a>.

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

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

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

%F Equals A001113 * A099286.

%F Equals A087197 * A125961.

%e 2.29069825230323823094953712686214731693708759053570...

%p evalf(exp(1)*erf(1), 120);  # _Alois P. Heinz_, Feb 10 2022

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

%o (PARI) exp(1)*(1 - erfc(1)) \\ _Michel Marcus_, Feb 10 2022

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

%K nonn,cons

%O 1,1

%A _Amiram Eldar_, Feb 10 2022