A260932 -id:A260932 - cp's OEIS frontend

A086053 Decimal expansion of Lengyel's constant L.

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

Offset: 1

Eric W. Weisstein, Jul 07 2003

L - log(Pi-1)/log(2) ~ 0.00000171037285384 ~ 1/Pi^11.5999410273. - Gerald McGarvey, Aug 17 2004

			1.0986858055251870130177463257213318079312220710644268407410427815783217...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 319 and 556.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
László Babai and Tamás Lengyel, A convergence criterion for recurrent sequences with application to the partition lattice, Analysis, Vol. 12, No. 1-2 (1992), pp. 109-120; preprint.
Tamás Lengyel, On a recurrence involving Stirling numbers, European Journal of Combinatorics, Vol. 5, No. 4 (1984), pp. 313-321.
Tamás Lengyel, On some 2-adic properties of a recurrence involving Stirling numbers, p-Adic Numbers Ultrametric Anal. Appl., Vol. 4, No. 3 (2012), pp. 179-186.
Simon Plouffe, The Lengyel constant.
Thomas Prellberg, On the asymptotic analysis of a class of linear recurrences (slides).
Eric Weisstein's World of Mathematics, Lengyel's Constant.

Cf. A005121, A131407, A330804, A331957.

Cf. A260932, A385521.

Equals lim_{n->oo} A005121(n) * (2*log(2))^n * n^(1+log(2)/3) / n!^2. - Amiram Eldar, Jun 27 2021

More terms from Vaclav Kotesovec, Mar 11 2014

A086555 E.g.f. satisfies F(x) = 1/2 * (F(-log(1-x)) + x).

Original entry on oeis.org

1, 1, 5, 47, 719, 16299, 513253, 21430513, 1145710573, 76317960163, 6197399680779, 602640663660199, 69134669061681469, 9239224408001877873, 1422887941494773642817, 250160794466824215921275

Offset: 1

Vladeta Jovovic, Sep 14 2003

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..250

Cf. A005121, A260932.

For a signed version see A246040.

Clear[a]; a[1] = 1; a[n_] := a[n] = Sum[Abs[StirlingS1[n, k]]*a[k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, May 29 2019 *)

a(n) = Sum_{k=1..n-1} |Stirling1(n, k)|*a(k).

a(n) ~ A260932 * n!^2 / (2^n * log(2)^n * n^(1 - log(2)/3)). - Vaclav Kotesovec, Jul 01 2025

A246040 a(1)=1; a(n)=Sum_{k=1..n-1} Stirling_1(n,k)*a(k).

Original entry on oeis.org

1, -1, 5, -47, 719, -16299, 513253, -21430513, 1145710573, -76317960163, 6197399680779, -602640663660199, 69134669061681469, -9239224408001877873, 1422887941494773642817, -250160794466824215921275, 49797413478450579190546203, -11142367835115998962269070519, 2784355004138005473128335461749

Offset: 1

N. J. A. Sloane, Aug 22 2014

sign

2*Sum_{k>=1} a(k-1)/fallfac(n,k) = -1/n + Sum_{k>=1} (1 + a(k-1))/n^k, with the falling factorials fallfac(n,k) = Product_{j=0..k-1}(n-j). - Vaclav Kotesovec, Aug 04 2015

D. Barsky, J.-P. Bézivin, p-adic Properties of Lengyel's Numbers, Journal of Integer Sequences, 17 (2014), #14.7.3. See Y_n.

A signed version of A086555.

Cf. A005121, A260932.

with(combinat);
Y:=proc(n) option remember; local k; if n=1 then 1 else add(stirling1(n,k)*Y(k),k=1..n-1); fi; end;
[seq(Y(n),n=1..35)];

Clear[a]; a[1] = 1; a[n_] := a[n] = Sum[StirlingS1[n, k]*a[k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Aug 04 2015 *)

a(n) ~ (-1)^(n+1) * c * n!^2 / (n^(1-log(2)/3) * (2*log(2))^n), where c = A260932 = 0.9031646749584662473216609915945142350500875792441051556... . - Vaclav Kotesovec, Aug 04 2015

A385521 Decimal expansion of a constant related to A375838.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jul 01 2025

Variant of Lengyel's constant A086053.

			1.59585433050036621247006569740016516964502505848324064247941890934119103861277...

Cf. A086053, A260932.
Cf. A086555, A246040, A375838.
Cf. A005121, A331957, A131407, A330804.

Equals lim_{n->oo} A375838(n) * 2^n * log(2)^n * n^(1-log(2)/3) / n!^2.

cp's OEIS Frontend

A086053 Decimal expansion of Lengyel's constant L.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A086555 E.g.f. satisfies F(x) = 1/2 * (F(-log(1-x)) + x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A246040 a(1)=1; a(n)=Sum_{k=1..n-1} Stirling_1(n,k)*a(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A385521 Decimal expansion of a constant related to A375838.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula