A092082 Triangle of numbers related to triangle A092083; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...
1, 7, 1, 91, 21, 1, 1729, 511, 42, 1, 43225, 15015, 1645, 70, 1, 1339975, 523705, 69300, 4025, 105, 1, 49579075, 21240765, 3226405, 230300, 8330, 147, 1, 2131900225, 984172735, 166428990, 13820205, 621810, 15386, 196, 1, 104463111025
Offset: 1
Examples
{1}; {7,1}; {91,21,1}; {1729,511,42,1}; ...
Links
- F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, (1992), pp. 24-48.
- P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem., arXiv:quant-phys/0402027, 2004.
- P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
- M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
- W. Lang, First 10 rows.
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
- Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - From N. J. A. Sloane, Aug 21 2012
Programs
-
Maple
# The function BellMatrix is defined in A264428. # Adds (1, 0, 0, 0, ..) as column 0. BellMatrix(n -> mul(6*k+1, k=0..n), 9); # Peter Luschny, Jan 26 2016
-
Mathematica
mmax = 9; a[n_, m_] := n!*Coefficient[Series[((-1 + (1 - 6*x)^(-1/6))^m)/m!, {x, 0, mmax}], x^n]; Flatten[Table[a[n, m], {n, 1, mmax}, {m, 1, n}]][[1 ;; 37]] (* Jean-François Alcover, Jun 22 2011, after e.g.f. *) rows = 9; t = Table[Product[6k+1, {k, 0, n}], {n, 0, rows}]; T[n_, k_] := BellY[n, k, t]; Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)
Formula
a(n, m) = sum(|A051151(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. with Wolfdieter Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.
a(n, m) = n!*A092083(n, m)/(m!*6^(n-m)); a(n+1, m) = (6*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n
E.g.f. for m-th column: ((-1+(1-6*x)^(-1/6))^m)/m!.
A092085 Alternating row sums of triangle A092082 (S2(7) Stirling2 generalization).
1, 6, 71, 1259, 29786, 881649, 31342599, 1300942894, 61754895021, 3299892654169, 196028028781586, 12815511452714431, 914381012515916401, 70704482046773444414, 5889818360271524031711, 525844048516367206170019
Offset: 0
Formula
a(n) = -Sum_{m=1..n} A092082(n, m)*(-1)^m, n >= 1.
E.g.f.: -(exp(1 - (1-6*x)^(-1/6)) - 1).
A132060 Row sums of triangle A132056 (S2(8), Stirling2 generalization).
1, 9, 145, 3361, 101601, 3786601, 167756689, 8611201665, 502522701121, 32854723149961, 2378687990172561, 188913154677765409, 16328988725252964385, 1525987835722540609641, 153312690475268723848081
Offset: 1
Comments
Generalized Bell numbers B(8,1;n).
Programs
-
Mathematica
With[{nn=30},Rest[CoefficientList[Series[Exp[-1+(1-7x)^(-1/7)]-1,{x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Sep 23 2011 *)
Formula
a(n)=sum(A132056(n, m), m=1..n), n>=1.
E.g.f.: exp(-1+(1-7*x)^(-1/7)) - 1.
Comments