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.

A094419 Generalized ordered Bell numbers Bo(6,n).

Original entry on oeis.org

1, 6, 78, 1518, 39390, 1277646, 49729758, 2258233998, 117196187550, 6842432930766, 443879517004638, 31674687990494478, 2465744921215207710, 207943837884583262286, 18885506918597311159518, 1837699347783655374914958, 190743171535070652261555870, 21035482423625416328497024206
Offset: 0

Views

Author

Ralf Stephan, May 02 2004

Keywords

Comments

Sixth row of array A094416, which has more information.

Links

Crossrefs

Cf. A032033, A094416, A094417, A094418, A131689.
Cf. A346985, A354252, A365555, A365556, A365557.
Cf. A384514, A384521, A384522, A384523.

Programs

  • Magma
    A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;
A094419:= func< k | A094416(6,k) >;
[A094419(n): n in [0..30]]; // G. C. Greubel, Jan 12 2024
  • Mathematica
    t = 30; Range[0, t]! CoefficientList[Series[1/(7 - 6 Exp[x]),{x, 0, t}], x] (* Vincenzo Librandi, Mar 16 2014 *)
  • PARI
    my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(7-6*exp(x)))) \\ Joerg Arndt, Jan 15 2024
  • PARI
    a(n) = (-1)^(n+1)*polylog(-n, 7/6)/7; \\ Seiichi Manyama, Jun 01 2025
  • SageMath
    def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
def A094419(k): return A094416(6,k)
[A094419(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

Formula

E.g.f.: 1/(7 - 6*exp(x)).
a(n) = Sum_{k=0..n} A131689(n,k) * 6^k. - Philippe Deléham, Nov 03 2008
a(n) ~ n! / (7*(log(7/6))^(n+1)). - Vaclav Kotesovec, Mar 14 2014
a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020
a(0) = 1; a(n) = 6 * a(n-1) - 7 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023
From Seiichi Manyama, Jun 01 2025: (Start)
a(n) = (-1)^(n+1)/7 * Li_{-n}(7/6), where Li_{n}(x) is the polylogarithm function.
a(n) = (1/7) * Sum_{k>=0} k^n * (6/7)^k.
a(n) = (6/7) * Sum_{k=0..n} 7^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)

A346985 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(1/6).

Original entry on oeis.org

1, 1, 8, 113, 2325, 62896, 2109143, 84403033, 3924963750, 207976793991, 12369246804853, 815880360117978, 59107920881218525, 4665585774576259261, 398534278371999103888, 36627974592437584634573, 3603954453161886215458025, 377983931878997401821759456, 42095013846928585982896180123
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 09 2021

Keywords

Comments

Stirling transform of A008542.
In general, for k >= 1, if e.g.f. = 1 / (k + 1 - k*exp(x))^(1/k), then a(n) ~ n! / (Gamma(1/k) * (k+1)^(1/k) * n^(1 - 1/k) * log(1 + 1/k)^(n + 1/k)). - Vaclav Kotesovec, Aug 14 2021

Links

Crossrefs

Cf. A000670, A008542, A094419, A305404, A346982, A346983, A346984, A352117, A352118, A352119.
Cf. A094419, A354252, A365555, A365556, A365557.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n<2, 1, (6*n-5)*g(n-1)) end:
b:= proc(n, m) option remember;
     `if`(n=0, g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 09 2021
  • Mathematica
    nmax = 18; CoefficientList[Series[1/(7 - 6 Exp[x])^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 18}]
  • Maxima
    a[n]:=if n=0 then 1 else (1/n)*sum(binomial(n,k)*(n+5*k)*a[k],k,0,n-1);
makelist(a[n],n,0,50); /* Tani Akinari, Aug 22 2023 */

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * A008542(k).
a(n) ~ n! / (Gamma(1/6) * 7^(1/6) * n^(5/6) * log(7/6)^(n + 1/6)). - Vaclav Kotesovec, Aug 14 2021
For n > 0, a(n) = (1/n)*Sum_{k=0..n-1} binomial(n,k)*(n+5*k)*a(k). - Tani Akinari, Aug 22 2023
O.g.f. (conjectural): 1/(1 - x/(1 - 7*x/(1 - 7*x/(1 - 14*x/(1 - 13*x/(1 - 21*x/(1 - ... - (6*n-5)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction). - Peter Bala, Aug 25 2023
a(0) = 1; a(n) = a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A354252 Expansion of e.g.f. 1/sqrt(7 - 6 * exp(x)).

Original entry on oeis.org

1, 3, 30, 489, 11127, 325218, 11612595, 489926559, 23846152332, 1315294430043, 81078316924035, 5523729981650004, 412148874577007037, 33425421047034028743, 2927620572178735480350, 275410244285003264624949, 27695140477706524122414867
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Links

Crossrefs

Cf. A305404, A354242, A354253.
Cf. A094419, A346985, A354252, A365556, A365557.
Cf. A011781.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(7-6*exp(x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(3*(exp(x)-1)/2)^k)))
  • PARI
    a(n) = sum(k=0, n, (3/2)^k*(2*k)!*stirling(n, k, 2)/k!);

Formula

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (3 * (exp(x) - 1)/2)^k.
a(n) = Sum_{k=0..n} (3/2)^k * (2*k)! * Stirling2(n,k)/k!.
a(n) ~ sqrt(2/7) * n^n / (exp(n) * log(7/6)^(n + 1/2)). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (6 - 3*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = 3*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A365556 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(2/3).

Original entry on oeis.org

1, 4, 44, 764, 18204, 551644, 20291804, 877970524, 43680345564, 2456429581404, 154072160204764, 10663000409493084, 807124301044917724, 66329628496719183964, 5881222650127663682524, 559616682597652939940444, 56879286407092006924382684
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2023

Keywords

Crossrefs

Cf. A094419, A346985, A354252, A365555, A365557.

Programs

  • Mathematica
    a[n_] := Sum[Product[6*j + 4, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 11 2023 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 6*j+4)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} 2^k * (Product_{j=0..k-1} (3*j+2)) * Stirling2(n,k) = Sum_{k=0..n} (Product_{j=0..k-1} (6*j+4)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (6 - 2*k/n) * binomial(n,k) * a(n-k).
O.g.f. (conjectural): 1/(1 - 4*x/(1 - 7*x/(1 - 10*x/(1 - 14*x/(1 - 16*x/(1 - 21*x/(1 - ... - (6*n - 2)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction). - Peter Bala, Sep 24 2023
a(n) ~ Gamma(1/3) * sqrt(3) * n^(n + 1/6) / (sqrt(2*Pi) * 7^(2/3) * exp(n) * log(7/6)^(n + 2/3)). - Vaclav Kotesovec, Nov 11 2023
a(0) = 1; a(n) = 4*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A365555 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(1/3).

Original entry on oeis.org

1, 2, 18, 274, 5938, 167122, 5786418, 237857874, 11319677618, 612109819602, 37069480301618, 2485356833141074, 182753029186750898, 14623552941626800082, 1265002802597606144818, 117633823750542653153874, 11701922865351577653913778
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2023

Keywords

Crossrefs

Cf. A094419, A346985, A354252, A365556, A365557.

Programs

  • Mathematica
    a[n_] := Sum[Product[6*j + 2, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 11 2023 *)
With[{nn=20},CoefficientList[Series[1/(7-6Exp[x])^(1/3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 21 2024 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 6*j+2)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} 2^k * (Product_{j=0..k-1} (3*j+1)) * Stirling2(n,k) = Sum_{k=0..n} (Product_{j=0..k-1} (6*j+2)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (6 - 4*k/n) * binomial(n,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n - 1/6) / (Gamma(1/3) * 7^(1/3) * exp(n) * log(7/6)^(n + 1/3)). - Vaclav Kotesovec, Sep 09 2023
From Peter Bala, Sep 24 2023: (Start)
O.g.f. (conjectural): 1/(1 - 2*x/(1 - 7*x/(1 - 8*x/(1 - 14*x/(1 - 14*x/(1 - 21*x/(1 - ... - (6*n - 4)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction).
More generally, it appears that the o.g.f. of the sequence whose e.g.f. is 1/(r+1 - r*exp(s*x))^(m/s) corresponds to the S-fraction 1/(1 - r*m*x/(1 - s*(r+1)*x/(1 - r*(m+s)*x/(1 - 2*s(r+1)*x/(1 - r*(m+2*s)*x/(1 - 3*s(r+1)*x/( 1 - ... ))))))). This is the case r = 6, s = 1, m = 1/3. (End)
a(0) = 1; a(n) = 2*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023
Showing 1-5 of 5 results.

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