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
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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
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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!.
A092086 Row sums of triangle A092083 (s2(7)).
1, 22, 589, 17158, 523930, 16486744, 529725541, 17282788798, 570508962718, 19007589409636, 638032097840818, 21549790120416700, 731641432814800132, 24950167895840374876, 854088761155288136341, 29334095386695889771054
Offset: 1
Crossrefs
Cf. A092083.
Programs
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Mathematica
Rest[CoefficientList[Series[(-1+(1-36x)^(-1/6))/(7-(1-36x)^(-1/6)), {x,0,20}],x]] (* Harvey P. Dale, Jun 29 2011 *)
Formula
a(n) = Sum_{m=1..n} A092083(n, m), n>=1.
G.f.: (-1 + (1-36*x)^(-1/6))/(7-(1-36*x)^(-1/6)).
A092087 Alternating row sums of triangle A092083 (s2(7)).
1, 20, 505, 14090, 415654, 12706406, 398165665, 12706132610, 411175121230, 13453601230544, 444162996339226, 14772945441872060, 494426375286105640, 16635957551869533770, 562327513989662942929, 19084061209362462745826
Offset: 0
Formula
a(n)= -sum(A092083(n, m)*(-1)^m, m=1..n), n>=1.
G.f.: (-1 +(1-36*x)^(-1/6))/(5+(1-36*x)^(-1/6)).
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.
A132058 Row sums of triangle A132057 (s2(8)).
1, 29, 1037, 40559, 1667583, 70782699, 3071608779, 135473190854, 6049729693582, 272822775416318, 12401578633961126, 567447248339504450, 26107796156861857866, 1206858263561650517658, 56014709781906608746434
Offset: 1
Formula
a(n)=sum(A132057(n, m), m=1..n), n>=1.
G.f.: (-1 + (1-49*x)^(-1/7))/(8-(1-49*x)^(-1/7)).
A132059 Alternating row sums of triangle A132057 (s2(8)).
1, 27, 925, 35069, 1406679, 58491537, 2493656187, 108280678092, 4768395658314, 212335592489544, 9540877059969102, 431908789303835976, 19675192863275361294, 901089855844979674068, 41459199062515242868098
Offset: 1
Formula
a(n)= -sum(A132057(n, m)*(-1)^m, m=1..n), n>=1.
G.f.: (-1 +(1-49*x)^(-1/7))/(6+(1-49*x)^(-1/7)).
Comments