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.

Previous Showing 71-76 of 76 results.

A036078 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=8.

Original entry on oeis.org

1, 2, 13, 127, 1508, 20859, 332557, 6019108, 121462267, 2692076295, 64846340130, 1684713690917, 46916754353013, 1393010598959594, 43889040801834505, 1461369418905803027, 51243270154712083052
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
  • T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Links

Crossrefs

Cf. A001861, A002872-A002875, A036074, ...

Programs

  • Mathematica
    mx = 16; p = 8; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 8^k * BellB[k, 1/8] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

Formula

a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=8. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (8*n/LambertW(8*n))^n * exp(n/LambertW(8*n) + (8*n/LambertW(8*n))^(1/8) - n - 9/8) / sqrt(1 + LambertW(8*n)). - Vaclav Kotesovec, Jul 10 2022

Extensions

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters.

A036079 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=9.

Original entry on oeis.org

1, 2, 14, 150, 1942, 29174, 505318, 9957798, 219177942, 5303780758, 139554619206, 3962202725254, 120644298135478, 3918518255860342, 135117086088186662, 4925731652244913766, 189170325211554345366, 7629758975467859662678, 322296334808561664346886
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
  • T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Links

Crossrefs

Cf. A001861, A002872, A002873, A002874, A002875, A036074...

Programs

  • Mathematica
    mx = 16; p = 9; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 9^k * BellB[k, 1/9] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

Formula

a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=9. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (9*n/LambertW(9*n))^n * exp(n/LambertW(9*n) + (9*n/LambertW(9*n))^(1/9) - n - 10/9) / sqrt(1 + LambertW(9*n)). - Vaclav Kotesovec, Jul 10 2022

Extensions

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters.

A036080 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=10.

Original entry on oeis.org

1, 2, 15, 175, 2452, 39703, 741177, 15771270, 375485507, 9837064575, 280338965720, 8623355105347, 284589703065137, 10022926411599482, 374900187362983015, 14830483377507515247, 618219446355189917804, 27071966121397255354079, 1241912851303663452150377
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
  • T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Links

Crossrefs

Cf. A001861, A002872, A002873, A002874, A002875, A036074, ...

Programs

  • Mathematica
    mx = 16; p = 10; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 10^k * BellB[k, 1/10] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

Formula

a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=10. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (10*n/LambertW(10*n))^n * exp(n/LambertW(10*n) + (10*n/LambertW(10*n))^(1/10) - n - 11/10) / sqrt(1 + LambertW(10*n)). - Vaclav Kotesovec, Jul 10 2022

Extensions

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters.

A036082 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=12.

Original entry on oeis.org

1, 2, 17, 231, 3724, 68819, 1464781, 35645040, 973624491, 29313919207, 960689482494, 33997330377817, 1291521482389621, 52395164853506674, 2259005857941805253, 103064324686839195035, 4957382457319437575820, 250592665906288206715951, 13275467282249493427541201
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

In general, for p>=2, a(n) ~ c * (p*n/LambertW(p*n))^n * exp(n/LambertW(p*n) + (p*n/LambertW(p*n))^(1/p) - n - 1 - 1/p) / sqrt(1 + LambertW(p*n)), where c = 1 for p>=3 and c = exp(-1/4) for p=2. - Vaclav Kotesovec, Jul 10 2022

References

  • T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
  • T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Links

Crossrefs

Cf. A001861, A002872, A002873, A002874, A002875, A036074, ...

Programs

  • Mathematica
    mx = 16; p = 12; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 12^k * BellB[k, 1/12] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

Formula

a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=12. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (12*n/LambertW(12*n))^n * exp(n/LambertW(12*n) + (12*n/LambertW(12*n))^(1/12) - n - 13/12) / sqrt(1 + LambertW(12*n)). - Vaclav Kotesovec, Jul 10 2022

Extensions

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters.

A292831 Expansion of 1 - 2*x - 2*x^2/(1 - 3*x - 4*x^2/(1 - 4*x - 6*x^2/(1 - 5*x - 8*x^2/(1 - 6*x - 10*x^2/(...))))), a continued fraction.

Original entry on oeis.org

1, -2, -2, -6, -26, -134, -778, -4950, -33946, -248230, -1921130, -15650358, -133644026, -1192354310, -11084816458, -107138260758, -1074526263898, -11163814083430, -119971275641642, -1331739929195766, -15250978417105082, -179975143242023366
Offset: 0

Views

Author

Seiichi Manyama, Sep 24 2017

Keywords

Examples

			G.f. = 1 - 2*x - 2*x^2 - 6*x^3 - 26*x^4 - 134*x^5 - 778*x^6 - 4950*x^7 - 33946*x^8 - ...

Crossrefs

Cf. A001861.

Formula

Convolution inverse of A001861.

A335501 a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} binomial(n,k)^2 * (n-k) * a(k).

Original entry on oeis.org

1, 2, 10, 86, 1098, 19142, 431926, 12150518, 414474570, 16781350694, 792845706630, 43107783435158, 2666346336398454, 185796230244565462, 14464057604306584774, 1248919312238777955086, 118855834572748011228490, 12397162719421869533115622
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 12 2020

Keywords

Crossrefs

Cf. A001861, A023998.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = (2/n) Sum[Binomial[n, k]^2 (n - k) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[2 Sum[x^k/(k!)^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

a(n) = (n!)^2 * [x^n] exp(2 * Sum_{k>=1} x^k / (k!)^2).
a(n) = (n!)^2 * [x^n] exp(2 * (BesselI(0,2*sqrt(x)) - 1)).
Previous Showing 71-76 of 76 results.

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

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