A094422 -id:A094422 - cp's OEIS frontend

A094418 Generalized ordered Bell numbers Bo(5,n).

1, 5, 55, 905, 19855, 544505, 17919055, 687978905, 30187495855, 1490155456505, 81732269223055, 4931150091426905, 324557348772511855, 23141780973332248505, 1776997406800302687055, 146197529083891406394905, 12829862285488250150167855, 1196280147496701351115120505

Offset: 0

Ralf Stephan, May 02 2004

nonn

Fifth row of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

Cf. A094416, A094417, A094419, A094422, A131689.

Cf. A346984, A365568, A365569, A365570.

A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;
A094418:= func< k | A094416(5,k) >;
[A094418(n): n in [0..30]]; // G. C. Greubel, Jan 12 2024

t = 30; Range[0, t]! CoefficientList[Series[1/(6 - 5 Exp[x]), {x, 0, t}], x] (* Vincenzo Librandi, Mar 16 2014 *)

my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(6 - 5*exp(x)))) \\ Joerg Arndt, Jan 15 2024

def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
def A094418(k): return A094416(5,k)
[A094418(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

E.g.f.: 1/(6 - 5*exp(x)).
a(n) = Sum_{k=0..n} A131689(n,k) * 5^k. - Philippe Deléham, Nov 03 2008
a(n) ~ n! / (6*(log(6/5))^(n+1)). - Vaclav Kotesovec, Mar 14 2014
a(0) = 1; a(n) = 5 * Sum_{k=1..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020
a(0) = 1; a(n) = 5 * a(n-1) - 6 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023
From Seiichi Manyama, Jun 01 2025: (Start)
a(n) = (-1)^(n+1)/6 * Li_{-n}(6/5), where Li_{n}(x) is the polylogarithm function.
a(n) = (1/6) * Sum_{k>=0} k^n * (5/6)^k.
a(n) = (5/6) * Sum_{k=0..n} 6^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)

A094416 Array read by antidiagonals: generalized ordered Bell numbers Bo(r,n).

Original entry on oeis.org

1, 2, 3, 3, 10, 13, 4, 21, 74, 75, 5, 36, 219, 730, 541, 6, 55, 484, 3045, 9002, 4683, 7, 78, 905, 8676, 52923, 133210, 47293, 8, 105, 1518, 19855, 194404, 1103781, 2299754, 545835, 9, 136, 2359, 39390, 544505, 5227236, 26857659, 45375130, 7087261

Offset: 1

Ralf Stephan, May 02 2004

Also, r times the number of (r+1)-level labeled linear rooted trees with n leaves.
"AIJ" (ordered, indistinct, labeled) transform of {r,r,r,...}.
Stirling transform of r^n*n!, i.e. of e.g.f. 1/(1-r*x).
Also, Bo(r,s) is ((x*d/dx)^n)(1/(1+r-r*x)) evaluated at x=1.
r-th ordered Bell polynomial (A019538) evaluated at n.
Bo(r,n) is the n-th moment of a geometric distribution with probability parameter = 1/(r+1).  Here, geometric distribution is the number of failures prior to the first success. - Geoffrey Critzer, Jan 01 2019
Row r (starting at r=0), Bo(r+1, n), is the Akiyama-Tanigawa algorithm applied to the powers of r+1. See Python program below. - Shel Kaphan, May 03 2024

			Array begins as:
  1,  3,   13,    75,     541,     4683,      47293, ...
  2, 10,   74,   730,    9002,   133210,    2299754, ...
  3, 21,  219,  3045,   52923,  1103781,   26857659, ...
  4, 36,  484,  8676,  194404,  5227236,  163978084, ...
  5, 55,  905, 19855,  544505, 17919055,  687978905, ...
  6, 78, 1518, 39390, 1277646, 49729758, 2258233998, ...

G. C. Greubel, Antidiagonals n = 1..50, flattened
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the log-normal distribution, arXiv:quant-ph/0303030, 2003.
C. G. Bower, Transforms

Rows 1-10 are A000670, A004123, A032033, A094417, A094418, A094419, A238464, A238465, A238466, A238467.
Columns include A014105, A094421.
Main diagonal is A094420.
Antidiagonal sums are A094422.
Cf. A019538, A131689, A344499.

A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;
[A094416(n-k+1,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jan 12 2024

Bo[, 0]=1; Bo[r, n_]:= Bo[r, n]= r*Sum[Binomial[n,k] Bo[r,n-k], {k, n}];
Table[Bo[r-n+1, n], {r, 10}, {n, r}] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

# The Akiyama-Tanigawa algorithm applied to the powers of r + 1
# generates the rows. Adds one row (r=0) and one column (n=0).
# Adapted from Peter Luschny on A371568.
def f(n, r): return (r + 1)**n
def ATtransform(r, len, f):
  A = [0] * len
  R = [0] * len
  for n in range(len):
      R[n] = f(n, r)
      for j in range(n, 0, -1):
          R[j - 1] = j * (R[j] - R[j - 1])
      A[n] = R[0]
  return A
for r in range(8): print([r], ATtransform(r, 8, f)) # Shel Kaphan, May 03 2024

def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
flatten([[A094416(n-k+1,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Jan 12 2024

E.g.f.: 1/(1 + r*(1 - exp(x))).
Bo(r, n) = Sum_{k=0..n} k!*r^k*Stirling2(n, k) = 1/(r+1) * Sum_{k>=1} k^n * (r/(r+1))^k, for r>0, n>0.
Recurrence: Bo(r, n) = r * Sum_{k=1..n} C(n, k)*Bo(r, n-k), with Bo(r, 0) = 1.
Bo(r,0) = 1, Bo(r,n) = r*Bo(r,n-1) - (r+1)*Sum_{j=1..n-1} (-1)^j * binomial(n-1,j) * Bo(r,n-j). - Seiichi Manyama, Nov 17 2023

Offset corrected by Geoffrey Critzer, Jan 01 2019

A372346 a(n) = Sum_{j=0..n} p(n - j, j) where p(n, x) = Sum_{k=0..n} k! * Stirling2(n, k) * x^k. Row sums of A344499.

Original entry on oeis.org

1, 1, 2, 6, 27, 175, 1532, 17276, 243093, 4165261, 85133686, 2039546786, 56447550543, 1783865468187, 63766726231792, 2558290237404920, 114418196763735113, 5670168958036693977, 309630356618418661738, 18536683645526372648446, 1211038603734731649106307, 85983731724631359047504967

Offset: 0

Peter Luschny, Apr 28 2024

nonn

Cf. A344499, A131689, A094422.

p := n -> local k; add(k!*Stirling2(n, k)*x^k, k = 0..n):
a := n -> local j; add(subs(x = j, p(n - j)),  j = 0..n):
seq(a(n), n = 0..21);

a(n) = A094422(n - 1) + 1.

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A094418 Generalized ordered Bell numbers Bo(5,n).

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A094416 Array read by antidiagonals: generalized ordered Bell numbers Bo(r,n).

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A372346 a(n) = Sum_{j=0..n} p(n - j, j) where p(n, x) = Sum_{k=0..n} k! * Stirling2(n, k) * x^k. Row sums of A344499.

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