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

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

Author

Ralf Stephan, May 02 2004

Keywords

• nonn
• tabl

Comments

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

Examples

			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, ...
		

Links

• 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

Crossrefs

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.

Programs

Magma

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

Mathematica

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 *)

Python

# 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

SageMath

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

Formula

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

Extensions

Offset corrected by Geoffrey Critzer, Jan 01 2019