A385521 -id:A385521 - 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

A260932 Decimal expansion of a constant related to A246040.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Aug 04 2015

Variant of Lengyel's constant A086053.

			0.9031646749584662473216609915945142350500875792441051556...

Cf. A246040, A086555.

Cf. A086053, A385521.

Equals lim_{n->oo} A246040(n) / ((-1)^(n+1)*(n!)^2/(n^(1-log(2)/3)*(2*log(2))^n)).

Equals lim_{n->oo} A086555(n) * 2^n * log(2)^n * n^(1 - log(2)/3) / n!^2. - Vaclav Kotesovec, Jul 01 2025

A375838 Number of rooted chains starting with the cycle (1)(2)(3)...(n) in the permutation poset of [n].

Original entry on oeis.org

1, 1, 2, 9, 83, 1270, 28799, 906899, 37866842, 2024422837, 134850653405, 10950546880152, 1064840930492393, 122158078221727119, 16325324374155336370, 2514183676808883419043, 442023695390488997377405, 87989953715757624724243004, 19688099473681895327628896249, 4919839221134662388853128069571, 1365091729320293490230304687026514

Offset: 0

Rajesh Kumar Mohapatra, Subhashree Sahoo, and Ranjan Kumar Dhani, Sep 10 2024

nonn

			Consider the set S = {1, 2, 3}. The a(3) = 1 + 5 + 3 = 9 in the poset of permutations of {1,2,3}:
 |{(1)(2)(3)}| = 1;
 |{(1)(2)(3) < (1)(23), (1)(2)(3) < (2)(13), (1)(2)(3) < (3)(12), (1)(2)(3) < (123), (1)(2)(3) < (132)}|=5;
 |{(1)(2)(3) < (1)(23) < (123), (1)(2)(3) < (2)(13)< (132), (1)(2)(3) < (3)(12) < (123)}| = 3.

Vaclav Kotesovec, Table of n, a(n) for n = 0..260

Row sums of A375837.
Cf. A048994, A331955, A330804, A331956, A331957, A375835, A375836.
Cf. A385521.

a:= proc(n) option remember;
      1+add(abs(Stirling1(n, k))*a(k), k=1..n-1)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 01 2025

T[n_, k_] := T[n, k] = If[k < 0 || k > n, 0, If[(n == 0 && k == 0) || k == 1, 1, Sum[If[r >= 0, Abs[StirlingS1[n, r]]*T[r, k - 1], 0], {r, k - 1, n - 1}]]]; Table[Sum[T[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 01 2025, after A375837 *)

a(n) = Sum_{k=0..n} A375837(n,k).
a(n) = (A375836(n)+1)/2.
Conjecture: a(n) = R(n,0) where R(n,k) = (k+1) * (Sum_{i=0..n-1} R(n-1,i) + Sum_{j=0..k-1} R(n-1,j)) for 0 <= k < n, R(n,n) = 1. - Mikhail Kurkov, Jun 21 2025
a(n) ~ c * n!^2 / (2^n * log(2)^n * n^(1-log(2)/3)), where c = A385521 = 1.59585433050036621247006569740016516964502505848324064247941890934119103861277... - Vaclav Kotesovec, Jul 01 2025
a(n) = 1 + Sum_{k=1..n-1} abs(Stirling1(n,k))*a(k). - Rajesh Kumar Mohapatra, Jul 01 2025

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A086053 Decimal expansion of Lengyel's constant L.

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A260932 Decimal expansion of a constant related to A246040.

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A375838 Number of rooted chains starting with the cycle (1)(2)(3)...(n) in the permutation poset of [n].

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