A049385 Triangle of numbers related to triangle A049375; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297...
1, 6, 1, 66, 18, 1, 1056, 372, 36, 1, 22176, 9240, 1200, 60, 1, 576576, 271656, 42840, 2940, 90, 1, 17873856, 9269568, 1685376, 142800, 6090, 126, 1, 643458816, 360847872, 73313856, 7254576, 386400, 11256, 168, 1, 26381811456, 15799069440
Offset: 1
Examples
Triangle begins:
{1};
{6,1};
{66,18,1};
{1056,372,36,1};
...
Links
- 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. Added Mar 01 2014.
- F. Bergeron, Philippe Flajolet and Bruno Salvy, Varieties of increasing trees, HAL, Rapport De Recherche Inria. Added Mar 01 2014.
- P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
- M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
- W. Lang, First 10 rows.
- Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO]. - From _N. J. A. Sloane_, Aug 21 2012
- E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's board, Discr. Maths. 239 (2001) 33-51.
Crossrefs
Cf. A049412.
Programs
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Maple
# The function BellMatrix is defined in A264428. # Adds (1,0,0,0, ..) as column 0. BellMatrix(n -> mul(5*k+1, k=0..n), 9); # Peter Luschny, Jan 28 2016
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Mathematica
a[n_, m_] := n!*Coefficient[Series[((-1 + (1 - 5*x)^(-1/5))^m)/m!, {x, 0, n}], x^n]; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 38]] (* Jean-François Alcover, Jun 21 2011, after e.g.f. *) rows = 9; t = Table[Product[5k+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) = n!*A049375(n, m)/(m!*5^(n-m)); a(n+1, m) = (5*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n
a(n, m) = sum(|A051150(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. to 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.
A039746 Row sums of triangle A049375.
1, 16, 306, 6321, 136311, 3021276, 68250216, 1563397131, 36195839771, 845073322136, 19864512260726, 469561887175791, 11151848709622581, 265910204875935096, 6362320059617532336, 152684022853771521426
Offset: 1
Keywords
Links
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Formula
G.f. (1-(1-25*x)^(1/5))/(6*(1-25*x)^(1/5)-1) = p(x)/(1-p(x)) where p(x)= (-1+(1-25*x)^(-1/5))/5 ( G.f. for first column of A049375).
A092083 A convolution triangle of numbers obtained from A034789.
1, 21, 1, 546, 42, 1, 15561, 1533, 63, 1, 466830, 54054, 2961, 84, 1, 14471730, 1885338, 124740, 4830, 105, 1, 458960580, 65542932, 4977882, 236880, 7140, 126, 1, 14801478705, 2277656901, 192582117, 10661301, 399735, 9891, 147, 1
Offset: 1
Comments
Examples
{1}; {21,1}; {546,42,1}; {15561,1533,63,1}; ...
Links
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
- W. Lang, First 10 rows.
Formula
a(n, m) = 6*(6*(n-1)+m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n
A132057 A convolution triangle of numbers obtained from A034904.
1, 28, 1, 980, 56, 1, 37730, 2744, 84, 1, 1531838, 130340, 5292, 112, 1, 64337196, 6136956, 299782, 8624, 140, 1, 2766499428, 288408120, 16120314, 568008, 12740, 168, 1, 121034349975, 13561837212, 841627332, 34401528, 956970, 17640, 196, 1
Offset: 1
Comments
Examples
{1}; {28,1}; {980,56,1}; (37730,2744,84,1);...
Links
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
- W. Lang, First 10 rows.
Programs
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Mathematica
a[n_, m_] := a[n, m] = 7*(7*(n-1) + m)*a[n-1, m]/n + m*a[n-1, m-1]/n; a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 8}, {m, 1, n}]][[1 ;; 36]] (* Jean-François Alcover, Jun 17 2011 *)
Formula
a(n, m) = 7*(7*(n-1)+m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n
G.f. for m-th column: ((-1+(1-49*x)^(-1/7))/7)^m.
Comments