A061684
Generalized Bell numbers: column 3 of A275043.
Original entry on oeis.org
1, 1, 5, 64, 1613, 69026, 4566992, 437665649, 57903766797, 10193400044254, 2319001344297830, 665816738235745559, 236563125351122920088, 102303284135845463907107, 53093636013475924370369829, 32666276100771741793923209939, 23573762287735885858839134983437
Offset: 0
Probably
A061698 from the same paper is an erroneous version of this sequence. -
Les Reid, Jan 01 2011
-
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-i)*binomial(n-1, i-1)/i!^2, i=1..n))
end:
a:= n-> b(n)*n!^2:
seq(a(n), n=0..20); # Alois P. Heinz, May 11 2016
-
b[n_] := b[n] = If[n==0, 1, Sum[b[n-i]*Binomial[n-1, i-1]/i!^2, {i, 1, n}]];
a[n_] := b[n]*n!^2;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 14 2017, after Alois P. Heinz *)
-
{a(n)=n!^3*polcoeff(exp(sum(m=1, n, x^m/m!^3)+x*O(x^n)), n)} /* Paul D. Hanna, Mar 15 2012 */
A061691
Triangle of generalized Stirling numbers.
Original entry on oeis.org
1, 1, 2, 1, 9, 6, 1, 34, 72, 24, 1, 125, 650, 600, 120, 1, 461, 5400, 10500, 5400, 720, 1, 1715, 43757, 161700, 161700, 52920, 5040, 1, 6434, 353192, 2361016, 4116000, 2493120, 564480, 40320, 1, 24309, 2862330, 33731208, 96960024, 97161120, 39372480, 6531840, 362880
Offset: 1
Triangle begins:
1;
1,2;
1,9,6;
1,34,72,24;
1,125,650,600,120;
...
T(4,2) = 34:
There are 7 partitions of the set {1,2,3,4} into 2 blocks. The four partitions {1,2,3}{4}, {1,2,4}{3}, {1,3,4}{2} and {2,3,4}{1} give rise to 4*4 = 16 uniform block permutations while the remaining 3 partitions {1,2}{3,4}, {1,3}{2,4} and {1,4}{2,3} give 2!*3*3 = 18 uniform block permutations : thus in total there are 16+18 = 34 block permutations between the set partitions of {1,2,3,4} into 2 blocks.
- Alois P. Heinz, Rows n = 1..141, flattened
- M. Aguiar and R. C. Orellana, The Hopf algebra of uniform block permutations, 17th International Conference on Formal Power Series and Algebraic Combinatorics, Taormina, July 2005.
- D. Aldous and P. Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. 36 (1999), 413-432.
- D. G. Fitzgerald, A presentation for the monoid of uniform block permutations, Bull. Austral. Math. Soc. 68 (2003), 317-324.
- A. T. Irish, F. Quitin, U. Madhow, and M. Rodwell, Achieving multiple degrees of freedom in long-range mm-wave MIMO channels using randomly distributed relays, 2014.
- I. P. Lankham, Patience Sorting and Its Generalizations, arXiv:0705.4524 [math.CO], 2007.
- J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
-
#A061691
#J = sum {n>=0} z^n/n!^2
J := BesselJ(0, 2*i*sqrt(z)):
G := exp(x*(J(z)-1)):
Gser := simplify(series(G, z = 0, 12)):
for n from 1 to 10 do
P[n] := n!^2*sort(coeff(Gser, z, n)) od:
for n from 1 to 10 do seq(coeff(P[n],x,k), k = 1..n) od;
# yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1,
add(x*b(n-i)*binomial(n, i)/i!, i=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i)/i!, i=1..n))(b(n)*n!):
seq(T(n), n=1..12); # Alois P. Heinz, Sep 10 2019
-
max = 9; g := Exp[x*(BesselI[0, 2*Sqrt[z]] - 1)]; gser = Series[g, {z, 0, max}, {x, 0, max}]; t[n_, k_] := n!^2*SeriesCoefficient[ gser // Normal, {z, 0, n}, {x, 0, k}]; Flatten[ Table[ t[n, k], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Apr 04 2012, after Maple *)
A061695
Generalized Bell numbers.
Original entry on oeis.org
1, 27, 864, 36000, 1944000, 133358400, 11379916800, 1185137049600, 148142131200000, 21908575180800000, 3785801791242240000, 756127866850836480000, 172901238886557941760000, 44887821634010234880000000, 13132894100921851576320000000, 4301460581188603786297344000000
Offset: 1
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