A090219 -id:A090219 - cp's OEIS frontend

A090220 Denominators used in A090219 to compute formula for column sequences of array A078741.

1, 1, 1, 1, 2, 10, 20, 70, 560, 1680, 2800, 30800, 369600, 800800, 11211200, 168168000, 448448000, 7623616000, 137225088000, 434546112000, 8690922240000, 182509367040000, 669201012480000, 15391623287040000, 369398958888960000

Offset: 1

Wolfdieter Lang, Dec 01 2003

The corresponding numerator sequence is N(n) := [1, 6, 3, 1, 3, 3, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] for n=1..26.

			The fifth (k=7) column of A078741 needs in A090219 the factor b(5) := N(5)/a(5)= 3/2.

a(n) = lcm(seq(denominator(a(n+2, m))), m=1..n)), with the a(n, m) formula of A090219(n, m) but without the 1/b(n-2) factor and lcm denotes the least common multiple of a set of numbers.

N(n) := gcd(seq(numerator(a(n+2, m))), m=1..n)), with the a(n, m) formula of A090219(n, m) but without the 1/b(n-2) factor and gcd denotes the greatest common divisor > 1 of a set of numbers.

A078741 Triangle of generalized Stirling numbers S_{3,3}(n,k) read by rows (n>=1, 3<=k<=3n).

Original entry on oeis.org

1, 6, 18, 9, 1, 36, 540, 1242, 882, 243, 27, 1, 216, 13608, 94284, 186876, 149580, 56808, 11025, 1107, 54, 1, 1296, 330480, 6148872, 28245672, 49658508, 41392620, 18428400, 4691412, 706833, 63375, 3285, 90, 1, 7776, 7954848, 380841264, 3762380016, 13062960720

Offset: 1

N. J. A. Sloane, Dec 21 2002

The sequence of row lengths for this array is [1,4,7,10,..]= A016777(n-1), n>=1.
The g.f. for the k-th column, (with leading zeros and k>=3) is G(k,x)= x^ceiling(k/3)*P(k,x)/product(1-fallfac(p,3)*x,p=3..k), with fallfac(n,m) := A008279(n,m) (falling factorials) and P(k,x) := sum(A089517(k,m)*x^m,m=0..kmax(k)), k>=3, with kmax(k) := A004523(k-3)= floor(2*(k-3)/3)= [0,0,1,2,2,3,4,4,5,...]. For the recurrence of the G(k,x) see A089517. Wolfdieter Lang, Dec 01 2003
For the computation of the k-th column sequence see A090219.
Codara et al., show that T(n,k) gives the number of k-colorings of the graph nK_3 (the disjoint union of n copies of the complete graph K_3). An example is given below. - Peter Bala, Aug 15 2013

			From _Peter Bala_, Aug 15 2013: (Start)
The table begins
n\k |   3     4     5      6      7     8     9   10  11  12
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
  1 |   1
  2 |   6    18     9      1
  3 |  36   540  1242    882    243    27     1
  4 | 216 13608 94284 186876 149580 56808 11025 1107  54   1
...
Graph coloring interpretation of T(2,3) = 6:
The graph 2K_3 is 2 copies of K_3, the complete graph on 3 vertices:
    o b      o e
   / \      / \
  o---o    o---o
  a   c    d   f
The six 3-colorings of 2K_3 are ad|be|cf, ad|bf|ce, ae|bd|cf, ae|bf|cd, af|bd|ce, and af|be|cd. (End)

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027,2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Codara, O. M. D’Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers arXiv:1308.1700v1 [cs.DM]
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
W. Lang, First 6 rows.

Row sums give A069223. Cf. A078739.
The column sequences (without leading zeros) are A000400 (powers of 6), 18*A089507, 9*A089518, A089519, etc.
A089504, A069223 (row sums), A090212 (alternating row sums).
A078740, A090214.

a[n_, k_] := (-1)^k*Sum[(-1)^p*((p-2)*(p-1)*p)^n*Binomial[k, p], {p, 3, k}]/k!; Table[a[n, k], {n, 1, 6}, {k, 3, 3*n}] // Flatten (* Jean-François Alcover, Dec 04 2013 *)

a(n, k) = (((-1)^k)/k!)*Sum_{p = 3..k} (-1)^p* binomial(k, p)*fallfac(p, 3)^n, with fallfac(p, 3) := A008279(p, 3) = p*(p-1)*(p-2); 3 <= k <= 3*n, n >= 1, else 0. From eq.(19) with r = 3 of the Blasiak et al. reference.
E^n = Sum_{k = 3..3*n} a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^3d^3/dx^3.
The row polynomials R(n,x) are given by the Dobinski-type formula R(n,x) = exp(-x)*Sum_{k >= 0} (k*(k-1)*(k-2))^n*x^k/k!. - Peter Bala, Aug 15 2013

A089518 Third column (k=5) of array A078741 ((3,3)-Stirling2) divided by 9.

Original entry on oeis.org

1, 138, 10476, 683208, 42315696, 2570768928, 155010407616, 9318969502848, 559578466388736, 33585275183251968, 2015370124337581056, 120928294183739148288, 7255843732407562776576, 435354129897768445943808

Offset: 0

Wolfdieter Lang, Dec 01 2003

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
Index entries for linear recurrences with constant coefficients, signature (90, -1944, 8640).

Cf. A089513 (third column of A089504), A089519, A090219.

G.f.: (1+48*x)/((1-3*2*1*x)*(1-4*3*2*x)*(1-5*4*3*x)).

a(n)= (10*(5*4*3)^n - 8*(4*3*2)^n + (3*2*1)^n)/3 = b(n) + 48*b(n-1), with b(n) := A089513(n).

A089519 Fourth column (k=6) of array A078741 ((3,3)-Stirling2).

Original entry on oeis.org

1, 882, 186876, 28245672, 3762380016, 474431543712, 58322293189056, 7082435837377152, 854925864902090496, 102893307861680404992, 12365333752840511118336, 1484928368468173355231232

Offset: 0

Wolfdieter Lang, Dec 01 2003

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

Cf. A089514, A089517-8, A090219.

G.f.: (1+672*x+14400*x^2)/((1-3*2*1*x)*(1-4*3*2*x)*(1-5*4*3*x)*(1-6*5*4*x)).

a(n)= 20*(6*5*4)^n -30*(5*4*3)^n + 12*(4*3*2)^n - (3*2*1)^n = b(n) + 672*b(n-1) + 14400*b(n-2), with b(n) := A089514(n).

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A090220 Denominators used in A090219 to compute formula for column sequences of array A078741.

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A078741 Triangle of generalized Stirling numbers S_{3,3}(n,k) read by rows (n>=1, 3<=k<=3n).

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A089518 Third column (k=5) of array A078741 ((3,3)-Stirling2) divided by 9.

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A089519 Fourth column (k=6) of array A078741 ((3,3)-Stirling2).

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