cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

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Author

N. J. A. Sloane, Aug 22 2014

Keywords

Comments

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

Links

Crossrefs

A signed version of A086555.
Cf. A005121, A260932.

Programs

  • Maple
    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)];
  • Mathematica
    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 *)

Formula

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

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

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Author

Vaclav Kotesovec, Aug 04 2015

Keywords

Comments

Variant of Lengyel's constant A086053.

Examples

			0.9031646749584662473216609915945142350500875792441051556...

Crossrefs

Cf. A246040, A086555.
Cf. A086053, A385521.

Formula

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

A308444 a(0) = 1; a(n) = Sum_{k=1..n} Stirling2(n,k)*a(n-k).

Original entry on oeis.org

1, 1, 2, 6, 27, 178, 1701, 23444, 464207, 13175526, 535353033, 31114680549, 2585577239479, 307143443783879, 52156058585285410, 12661558539485464967, 4394996515200407462730, 2181761307828685811029286, 1549298114199282873678255787, 1574165879361329032738370945407
Offset: 0

Views

Author

Ilya Gutkovskiy, May 27 2019

Keywords

Links

Crossrefs

Cf. A005121, A008277, A086555, A246040.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-j)*Stirling2(n, j), j=1..n))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Feb 25 2025
  • Mathematica
    a[n_] := a[n] = Sum[StirlingS2[n, k] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]

Formula

log(a(n)) ~ n^2 * log(3) / 6. - Vaclav Kotesovec, May 28 2019

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

Views

Author

Vaclav Kotesovec, Jul 01 2025

Keywords

Comments

Variant of Lengyel's constant A086053.

Examples

			1.59585433050036621247006569740016516964502505848324064247941890934119103861277...

Crossrefs

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

Formula

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

A329968 a(0) = 1; a(n) = Sum_{k=1..n} |Stirling1(n,k)| * a(n-k).

Original entry on oeis.org

1, 1, 2, 8, 77, 2329, 302564, 222085736, 1123297137786, 45315713537365706, 16445319538981321524068, 59677257201788875416461684008, 2382127122661172512076372104185900762, 1141042791864963721866517729601372480006639394, 7105297245805890887235402087045369387823693986873653108
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 26 2019

Keywords

Links

Crossrefs

Cf. A008275, A086555, A308444, A329967.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-j)*abs(Stirling1(n, j)), j=1..n))
    end:
seq(a(n), n=0..14);  # Alois P. Heinz, Feb 25 2025
  • Mathematica
    a[n_] := a[n] = Sum[Abs[StirlingS1[n, k]] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 13}]
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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