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 1-6 of 6 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)

A326323 A(n, k) = A_{n}(k) where A_{n}(x) are the Eulerian polynomials, square array read by ascending antidiagonals, for n >= 0 and k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 13, 24, 1, 1, 1, 5, 22, 75, 120, 1, 1, 1, 6, 33, 160, 541, 720, 1, 1, 1, 7, 46, 285, 1456, 4683, 5040, 1, 1, 1, 8, 61, 456, 3081, 15904, 47293, 40320, 1, 1, 1, 9, 78, 679, 5656, 40005, 202672, 545835, 362880, 1
Offset: 0

Views

Author

Peter Luschny, Jun 27 2019

Keywords

Examples

			Array starts:
  k=0: 1, 1, 1,  1,    1,     1,      1,        1,         1, ... [A000012]
  k=1: 1, 1, 2,  6,   24,   120,    720,     5040,     40320, ... [A000142]
  k=2: 1, 1, 3, 13,   75,   541,   4683,    47293,    545835, ... [A000670]
  k=3: 1, 1, 4, 22,  160,  1456,  15904,   202672,   2951680, ... [A122704]
  k=4: 1, 1, 5, 33,  285,  3081,  40005,   606033,  10491885, ... [A255927]
  k=5: 1, 1, 6, 46,  456,  5656,  84336,  1467376,  29175936, ... [A326324]
  k=6: 1, 1, 7, 61,  679,  9445, 158095,  3088765,  68958295, ... [A384525]
  k=7: 1, 1, 8, 78,  960, 14736, 272448,  5881968, 145105920, ... [A384514]
  k=8: 1, 1, 9, 97, 1305, 21841, 440649, 10386817, 279768825, ...
Seen as a triangle:
  [0], 1
  [1], 1, 1
  [2], 1, 1, 1
  [3], 1, 1, 2,  1
  [4], 1, 1, 3,  6,   1
  [5], 1, 1, 4, 13,  24,    1
  [6], 1, 1, 5, 22,  75,  120,     1
  [7], 1, 1, 6, 33, 160,  541,   720,     1
  [8], 1, 1, 7, 46, 285, 1456,  4683,  5040,     1
  [9], 1, 1, 8, 61, 456, 3081, 15904, 47293, 40320, 1

Links

Crossrefs

Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324, A384525, A384514.

Programs

  • Maple
    A := (n, k) -> add(combinat:-eulerian1(k, j)*n^j, j=0..k):
seq(seq(A(n-k, k), k=0..n), n=0..10);
# Alternative:
egf := n -> `if`(n=1, 1/(1-x), (n-1)/(n  - exp((n-1)*x))):
ser := n -> series(egf(n), x, 21):
for n from 0 to 6 do seq(k!*coeff(ser(n), x, k), k=0..9) od;
  • Mathematica
    a[n_, 0] := 1; a[n_, 1] := n!;
a[n_, k_] := (k - 1)^(n + 1)/k HurwitzLerchPhi[1/k, -n, 0];
(* Alternative: *) a[n_, k_] := Sum[StirlingS2[n, j] (k - 1)^(n - j) j!, {j, 0, n}];
Table[Print[Table[a[n, k], {n, 0, 10}]], {k, 0, 8}]

Formula

A(n, k) = Sum_{j=0..k} a(k, j)*n^j where a(k, j) are the Eulerian numbers.
E.g.f.: (n - 1)/(n - exp((n-1)*x)) for n = 0 and n >= 2, 1/(1 - x) if n = 1.
A(n, 0) = 1; A(n, 1) = n!.
A(n, k) = (k - 1)^(n + 1)/k HurwitzLerchPhi(1/k, -n, 0) for k >= 2.
A(n, k) = Sum_{j=0..n} j! * Stirling2(n, j) * (k - 1)^(n - j) for k >= 2.

A384521 Expansion of e.g.f. 5/(7 - 2*exp(5*x)).

Original entry on oeis.org

1, 2, 18, 218, 3474, 69290, 1659330, 46359770, 1480241970, 53171142410, 2122154748450, 93168872862650, 4462242691496850, 231524863130863850, 12936797161953970050, 774495903492069700250, 49458416187322116299250, 3355754824852804221058250, 241081466990843266748993250
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2025

Keywords

Links

Crossrefs

Cf. A094419, A384514, A384522, A384523, A384524.

Programs

  • PARI
    a(n) = (-5)^(n+1)*polylog(-n, 7/2)/7;

Formula

a(n) = (-5)^(n+1)/7 * Li_{-n}(7/2), where Li_{n}(x) is the polylogarithm function.
a(n) = 5^(n+1)/7 * Sum_{k>=0} k^n * (2/7)^k.
a(n) = Sum_{k=0..n} 2^k * 5^(n-k) * k! * Stirling2(n,k).
a(n) = (2/7) * Sum_{k=0..n} 7^k * (-5)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 2 * Sum_{k=1..n} 5^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 2 * a(n-1) + 7 * Sum_{k=1..n-1} (-5)^(k-1) * binomial(n-1,k) * a(n-k).

A384522 Expansion of e.g.f. 4/(7 - 3*exp(4*x)).

Original entry on oeis.org

1, 3, 30, 426, 8040, 189768, 5375280, 177632976, 6708685440, 285038686848, 13456362881280, 698786099602176, 39586707755811840, 2429498408440009728, 160571526535426529280, 11370607719608891467776, 858870213271187908362240, 68928740686010010238353408
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2025

Keywords

Links

Crossrefs

Cf. A094419, A384514, A384521, A384523, A384524.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[4/(7-3Exp[4x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 21 2025 *)
  • PARI
    a(n) = (-4)^(n+1)*polylog(-n, 7/3)/7;

Formula

a(n) = (-4)^(n+1)/7 * Li_{-n}(7/3), where Li_{n}(x) is the polylogarithm function.
a(n) = 4^(n+1)/7 * Sum_{k>=0} k^n * (3/7)^k.
a(n) = Sum_{k=0..n} 3^k * 4^(n-k) * k! * Stirling2(n,k).
a(n) = (3/7) * Sum_{k=0..n} 7^k * (-4)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} 4^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3 * a(n-1) + 7 * Sum_{k=1..n-1} (-4)^(k-1) * binomial(n-1,k) * a(n-k).

A384523 Expansion of e.g.f. 3/(7 - 4*exp(3*x)).

Original entry on oeis.org

1, 4, 44, 708, 15180, 406884, 13087404, 491114628, 21062220300, 1016197112484, 54476506976364, 3212426755972548, 206654933095516620, 14401921040252826084, 1080885666078491553324, 86916516692600836638468, 7455102038197447378720140, 679412933203279242481083684
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2025

Keywords

Links

Crossrefs

Cf. A094419, A384514, A384521, A384522, A384524.

Programs

  • PARI
    a(n) = (-3)^(n+1)*polylog(-n, 7/4)/7;

Formula

a(n) = (-3)^(n+1)/7 * Li_{-n}(7/4), where Li_{n}(x) is the polylogarithm function.
a(n) = 3^(n+1)/7 * Sum_{k>=0} k^n * (4/7)^k.
a(n) = Sum_{k=0..n} 4^k * 3^(n-k) * k! * Stirling2(n,k).
a(n) = (4/7) * Sum_{k=0..n} 7^k * (-3)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 4 * Sum_{k=1..n} 3^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4 * a(n-1) + 7 * Sum_{k=1..n-1} (-3)^(k-1) * binomial(n-1,k) * a(n-k).

A384524 Expansion of e.g.f. 2/(7 - 5*exp(2*x)).

Original entry on oeis.org

1, 5, 60, 1070, 25440, 756080, 26964960, 1121963120, 53351831040, 2854122433280, 169649803023360, 11092432778385920, 791204615734640640, 61138238969353748480, 5087702653663698677760, 453621615686933964830720, 43141424825262182799114240, 4359374368561019960377671680
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2025

Keywords

Links

Crossrefs

Cf. A094419, A384514, A384521, A384522, A384523.

Programs

  • Maple
    A384524:=proc(n)
    add(5^k * 2^(n-k) * k! * combinat[stirling2](n,k) ,k=0..n) ;
end proc:
seq(A384524(n), n=0..40); # R. J. Mathar, Jun 04 2025
  • PARI
    a(n) = (-2)^(n+1)*polylog(-n, 7/5)/7;

Formula

a(n) = (-2)^(n+1)/7 * Li_{-n}(7/5), where Li_{n}(x) is the polylogarithm function.
a(n) = 2^(n+1)/7 * Sum_{k>=0} k^n * (5/7)^k.
a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * k! * Stirling2(n,k).
a(n) = (5/7) * Sum_{k=0..n} 7^k * (-2)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 5 * Sum_{k=1..n} 2^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 5 * a(n-1) + 7 * Sum_{k=1..n-1} (-2)^(k-1) * binomial(n-1,k) * a(n-k).
Showing 1-6 of 6 results.

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