A035529 -id:A035529 - cp's OEIS frontend

A035469 Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.

1, 4, 1, 28, 12, 1, 280, 160, 24, 1, 3640, 2520, 520, 40, 1, 58240, 46480, 11880, 1280, 60, 1, 1106560, 987840, 295960, 40040, 2660, 84, 1, 24344320, 23826880, 8090880, 1296960, 109200, 4928, 112, 1, 608608000, 643843200

Offset: 1

Wolfdieter Lang

Previous name was: Triangle of numbers related to triangle A035529; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297 and A035342.
a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing quartic (4-ary) trees. Proof based on the a(n,m) recurrence. See a D. Callan comment on the m=1 case A007559. See also the F. Bergeron et al. reference, especially Table 1, first row and Example 1 for the e.g.f. for m=1. - Wolfdieter Lang, Sep 14 2007
For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle starts:
     {1}
     {4,    1}
    {28,   12,    1}
   {280,  160,   24,    1}
  {3640, 2520,  520,   40,    1}

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.

Peter Bala, Generalized Dobinski formulas
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Richell O. Celeste, Roberto B. Corcino and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Tom Copeland, A Class of Differential Operators and the Stirling Numbers
Tom Copeland, Mathemagical Forests
Tom Copeland, Addendum to Mathemagical Forests
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First 10 rows.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
E. Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 (2001) 33-51.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.

a(n, m)=: S2(4, n, m) is the fourth triangle of numbers in the sequence S2(1, n, m) := A008277(n, m) (Stirling 2nd kind), S2(2, n, m) := A008297(n, m) (Lah), S2(3, n, m) := A035342(n, m). a(n, 1)= A007559(n).
Row sums: A049119(n), n >= 1.
Cf. A094638.

a[n_, m_] /; n >= m >= 1 := a[n, m] = (3(n-1) + m)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* _Jean-François Alcover, Jul 22 2011 *)
rows = 9;
a[n_, m_] := BellY[n, m, Table[Product[3k+1, {k, 0, j}], {j, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: A007559(n+1) , 9) # Peter Luschny, Jan 19 2016

a(n, m) = Sum_{j=m..n} |A051141(n, j)|*S2(j, m) (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.
a(n, m) = n!*A035529(n, m)/(m!*3^(n-m)); a(n+1, m) = (3*n+m)*a(n, m) + a(n, m-1), n >= m >= 1; a(n, m) := 0, n < m; a(n, 0) := 0, a(1, 1)=1;
E.g.f. of m-th column: ((-1+(1-3*x)^(-1/3))^m)/m!.
From Peter Bala, Nov 25 2011: (Start)
E.g.f.: G(x,t) = exp(t*A(x)) = 1 + t*x + (4*t+t^2)*x^2/2! + (28*t + 12*t^2 + t^3)*x^3/3! + ..., where A(x) = -1 + (1-3*x)^(-1/3) satisfies the autonomous differential equation A'(x) = (1+A(x))^4.
The generating function G(x,t) satisfies the partial differential equation t*(dG/dt+G) = (1-3*x)*dG/dx, from which follows the recurrence given above.
The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^4*d/dx. Cf. A008277 (D = (1+x)*d/dx), A105278 (D = (1+x)^2*d/dx), A035342 (D = (1+x)^3*d/dx) and A049029 (D = (1+x)^5*d/dx).
(End)
Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k>=0} k*(k+3)*(k+6)*...*(k+3*(n-1))*x^k/k!. - Peter Bala, Jun 23 2014

New name from Peter Luschny, Jan 19 2016

A034171 Related to triple factorial numbers A007559(n+1).

Original entry on oeis.org

1, 6, 42, 315, 2457, 19656, 160056, 1320462, 11003850, 92432340, 781473420, 6642524070, 56716936290, 486145168200, 4180848446520, 36059817851235, 311811366125385, 2702365173086670, 23467908082068450, 204170800313995515, 1779202688450532345, 15527587099204645920

Offset: 0

Wolfdieter Lang

Working with an offset of 1, we conjecture a(p*n) = a(n) (mod p^2) for prime p = 1 (mod 3) and all positive integers n except those n of the form n = m*p + k for 0 <= m <= (p-1)/3 and 1 <= k <= (p-1)/3. Cf. A298799, A004981 and A004982. - Peter Bala, Dec 23 2019

Michael De Vlieger, Table of n, a(n) for n = 0..1050
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A007559, A025748, A034164, A035529.

Cf. A298799, A004981, A004982, A004987, A073005.

CoefficientList[Series[(-1 + (1 - 9 x)^(-1/3))/(3 x), {x, 0, 19}], x] (* Michael De Vlieger, Oct 13 2019 *)

a(n) = 3^n*A007559(n+1)/(n+1)! where A007559(n+1)=(3*n+1)!!!.
G.f.: (-1+(1-9*x)^(-1/3))/(3*x).
a(n) = A035529(n+1, 1) (first column of triangle).
Convolution of A004987(n) with A025748(n+1), n >= 0.
From R. J. Mathar, Jan 28 2020: (Start)
D-finite with recurrence: (n+1)*a(n) + 3*(-3*n-1)*a(n-1) = 0.
G.f.: (1F0(1/3;;9*x)-1)/(3*x). (End)
Sum_{n>=0} 1/a(n) = 3/8 + 3*sqrt(3)*Pi/32 + 9*log(3)/32. - Amiram Eldar, Dec 22 2022
a(n) ~ 3^(2*n+1) * n^(-2/3) / Gamma(1/3). - Amiram Eldar, Aug 19 2025

A048882 A convolution triangle of numbers obtained from A034255.

Original entry on oeis.org

1, 10, 1, 120, 20, 1, 1560, 340, 30, 1, 21216, 5520, 660, 40, 1, 297024, 88032, 12880, 1080, 50, 1, 4243200, 1392768, 236448, 24640, 1600, 60, 1, 61526400, 21952320, 4187232, 512464, 41800, 2220, 70, 1, 902387200, 345396480, 72452160, 10060416

Offset: 0

Wolfdieter Lang

a(n,m)=: s2(5; n,m), generalizing s2(2; n,m) := A007318(n-1,m-1) (Pascal), s2(3; n,m) := A035324(n,m), s2(4; n,m)= A035529.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A035529, A034255. Row sums: A048965(n), n >= 1.

a(n+1, m) = 4*(4*n+m)*a(n, m)/(n+1) + m*a(n, m-1)/(n+1), n >= m >= 1; a(n, m) := 0, n

G.f. for column m: ((-1+(1-16*x)^(-1/4))/4)^m.

A092083 A convolution triangle of numbers obtained from A034789.

Original entry on oeis.org

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

Wolfdieter Lang, Mar 19 2004

a(n,1) = A034789(n). a(n,m)=: s2(7; n,m), a member of a sequence of unsigned triangles including s2(2; n,m)=A007318(n-1,m-1) (Pascal's triangle). s2(3; n,m)= A035324(n,m), s2(4; n,m)= A035529(n,m), s2(5; n,m)= A048882(n,m), s2(6; n,m)= A049375.

			{1}; {21,1}; {546,42,1}; {15561,1533,63,1}; ...

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First 10 rows.

Cf. A092086 (row sums), A092087 (alternating row sums).

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

A049375 A convolution triangle of numbers obtained from A034687.

Original entry on oeis.org

1, 15, 1, 275, 30, 1, 5500, 775, 45, 1, 115500, 19250, 1500, 60, 1, 2502500, 471625, 44625, 2450, 75, 1, 55412500, 11495000, 1254000, 85000, 3625, 90, 1, 1246781250, 279675000, 34093125, 2698875, 143750, 5025, 105, 1, 28398906250, 6802812500

Offset: 1

Wolfdieter Lang

a(n,1) = A034687(n). a(n,m)=: s2(6; n,m), a member of a sequence of unsigned triangles including s2(2; n,m)= A007318(n-1,m-1) (Pascal's triangle). s2(3; n,m)= A035324(n,m), s2(4; n,m)= A035529(n,m), s2(5; n,m)= A048882(n,m).

			{1}; {15,1}; {275,30,1}; {5500,775,45,1}; ...

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A039746.

a[n_, m_] := Coefficient[Series[((-1 + (1 - 25*x)^(-1/5))/5)^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 g.f. *)

a(n, m) = 5*(5*(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+(1-25*x)^(-1/5))/5)^m.

A132057 A convolution triangle of numbers obtained from A034904.

Original entry on oeis.org

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

Wolfdieter Lang Sep 14 2007

a(n,1) = A034904(n). a(n,m)=: s2(8; n,m), a member of a sequence of unsigned triangles including s2(2; n,m)=A007318(n-1,m-1) (Pascal's triangle). s2(3;n,m)= A035324(n,m), s2(4; n,m)= A035529(n,m), s2(5; n,m)= A048882(n,m), s2(6; n,m)= A049375; s2(7; n,m)=A092083.

			{1}; {28,1}; {980,56,1}; (37730,2744,84,1);...

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First 10 rows.

Cf. A132058 (row sums), A132059 (negative of alternating row sums).

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 *)

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.

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A049028 Row sums of triangle A035529.

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A035469 Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.

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A034171 Related to triple factorial numbers A007559(n+1).

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A048882 A convolution triangle of numbers obtained from A034255.

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A092083 A convolution triangle of numbers obtained from A034789.

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A049375 A convolution triangle of numbers obtained from A034687.

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A132057 A convolution triangle of numbers obtained from A034904.

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