# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a132056 Showing 1-1 of 1 %I A132056 #25 Aug 28 2019 15:44:17 %S A132056 1,8,1,120,24,1,2640,672,48,1,76560,22800,2160,80,1,2756160,920160, %T A132056 104880,5280,120,1,118514880,43243200,5639760,347760,10920,168,1, %U A132056 5925744000,2323918080,336510720,24071040,937440,20160,224,1 %N A132056 Triangle read by rows, the Bell transform of Product_{k=0..n} 7*k+1 without column 0. %C A132056 Previous name was: Triangle of numbers related to triangle A132057; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ... %C A132056 a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing 8-ary trees. See the F. Bergeron et al. reference, especially Table 1, first row, for the e.g.f. for m=1. %C A132056 a(n,m) := S2(8; n,m) is the eighth triangle of numbers in the sequence S2(k;n,m), k=1..7: A008277 (unsigned Stirling 2nd kind), A008297 (unsigned Lah), A035342, A035469, A049029, A049385, A092082, respectively. a(n,1)=A045754(n), n>=1. %H A132056 F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48. %H A132056 P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205. %H A132056 P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004. %H A132056 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. %H A132056 M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3 %H A132056 W. Lang, First 10 rows. %F A132056 a(n, m) = n!*A132057(n, m)/(m!*7^(n-m)); a(n+1, m) = (7*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n mul(7*k+1, k=0..n), 8); # _Peter Luschny_, Jan 27 2016 %t A132056 a[n_, m_] := a[n, m] = ((m*a[n-1, m-1]*(m-1)! + (m+7*n-7)*a[n-1, m]*m!)*n!)/(n*m!*(n-1)!); %t A132056 a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1; %t A132056 Flatten[Table[a[n, m], {n, 1, 8}, {m, 1, n}]][[1 ;; 36]] %t A132056 (* _Jean-François Alcover_, Jun 17 2011 *) %t A132056 rows = 8; %t A132056 a[n_, m_] := BellY[n, m, Table[Product[7k+1, {k, 0, j}], {j, 0, rows}]]; %t A132056 Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018 *) %Y A132056 Cf. A132060 (row sums), A132061 (alternating row sums). %Y A132056 Cf. A092082 S2(7) triangle. %K A132056 nonn,easy,tabl %O A132056 1,2 %A A132056 _Wolfdieter Lang_ Sep 14 2007 %E A132056 New name from _Peter Luschny_, Jan 27 2016 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE