A049213 -id:A049213 - cp's OEIS frontend

A000369 Triangle of numbers related to triangle A049213; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.

1, 3, 1, 21, 9, 1, 231, 111, 18, 1, 3465, 1785, 345, 30, 1, 65835, 35595, 7650, 825, 45, 1, 1514205, 848925, 196245, 24150, 1680, 63, 1, 40883535, 23586255, 5755050, 775845, 62790, 3066, 84, 1, 1267389585, 748471185, 190482705, 27478710

Offset: 1

Wolfdieter Lang

a(n,m) := S2p(-3; n,m), a member of a sequence of triangles including S2p(-1; n,m) := A001497(n-1,m-1) (Bessel triangle) and ((-1)^(n-m))*S2p(1; n,m) := A008277(n,m) (Stirling 2nd kind). a(n,1)= A008545(n-1).
a(n,m), n>=m>=1, enumerates unordered n-vertex m-forests composed of m increasing plane (aka ordered) trees, with one vertex of out-degree r=0 (leafs or a root) and each vertex with out-degree r>=1 comes in r+2 types (like for an (r+2)-ary vertex). Proof from the e.g.f. of the first column Y(z):=1-(1-4*x)^(1/4) and the F. Bergeron et al. reference given in A001498, eq. (8), Y'(z)= phi(Y(z)), Y(0)=0, with out-degree o.g.f. phi(w)=1/(1-w)^3. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+3) (A008545) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015

			Triangle begins:
  1;
  3, 1;
  21, 9, 1;
  231, 111, 18, 1;
  3465, 1785, 345, 30, 1; ...
Tree combinatorics for a(3,2)=9: there are three m=2 forests each with one tree a root (with out-degree r=0) and the other tree a root and a leaf coming in three versions (like for a 3-ary vertex). Each such forest can be labeled increasingly in three ways (like (1,(23)), (2,(13)) and (3,(12))) yielding 9 such forests. - _Wolfdieter Lang_, Oct 12 2007

Vincenzo Librandi, Rows n = 1..50, flattened
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0212072, 2002.
Tom Copeland, A Class of Differential Operators and the Stirling Numbers
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
Wolfdieter Lang, First ten rows.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
Mathias Pétréolle, Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
Index entries for sequences related to Bessel functions or polynomials

Row sums give A016036. Cf. A004747.
Columns include A008545.
Alternating row sums A132163.

a[n_, m_] /; n >= m >= 1 := a[n, m] = (4(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 *)

# uses[bell_transform from A264428]
# Adds a column 1,0,0,0,... at the left side of the triangle.
def A000369_row(n):
    multifact_4_3 = lambda n: prod(4*k + 3 for k in (0..n-1))
    mfact = [multifact_4_3(k) for k in (0..n)]
    return bell_transform(n, mfact)
[A000369_row(n) for n in (0..9)] # Peter Luschny, Dec 31 2015

a(n, m) = n!*A049213(n, m)/(m!*4^(n-m)); a(n+1, m) = (4*n-m)*a(n, m) + a(n, m-1), n >= m >= 1; a(n, m) := 0, n

E.g.f. of m-th column: ((1-(1-4*x)^(1/4))^m)/m!.

From Peter Bala, Jun 08 2016: (Start)

With offset 0, the e.g.f. is 1/(1 - 4*x)^(3/4)*exp(t*(1 - (1 - 4*x)^(1/4))) = 1 + (3 + t)*x + (21 + 9*t + t^2)*x^2/2! + ....

Thus with row and column numbering starting at 0, this triangle is the exponential Riordan array [d/dx(F(x)), F(x)], belonging to the Derivative subgroup of the exponential Riordan group, where F(x) = 1 - (1 - 4*x)^(1/4).

Row polynomial recurrence: R(n+1,t) = t*Sum_{k = 0..n} binomial(n,k)*A008545(k)*R(n-k,t) with R(0,t) = 1. (End)

A025757 4th-order Vatalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 7, 69, 783, 9597, 123495, 1643397, 22413183, 311466829, 4392857431, 62702224213, 903886452975, 13138698859677, 192337495360071, 2832859169364261, 41946319269028191, 624009420903043821

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
T. M. Richardson, The Super Patalan Numbers, arXiv preprint arXiv:1410.5880, 2014
T. M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015) # 15.3.3.

a(n), n >= 1, = row sums of triangle A049213.

Cf. A000108, A025756, A025758, A025759, A025760, A025761, A025762, A025763.

Table[SeriesCoefficient[4/(3 + (1 - 16*x)^(1/4)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)

G.f.: 4 / (3+(1-16*x)^(1/4)).
a(n) = Sum_{m=1..n-1} (m/n*4^(n-m)) * Sum_{k=1..n-m} binomial(n+k-1,n-1) * Sum_{j=0..k} binomial(j,n-m-3*k+2*j) * 4^(j-k) * binomial(k,j) * 3^(-n+m+3*k-j) * 2^(n-m-3*k+j) * (-1)^(n-m-3*k+2*j) + 1. - Vladimir Kruchinin, Feb 08 2011
Conjecture: 5*n*(n-1)*(n-2)*a(n) -(239*n-600)*(n-1)*(n-2)*a(n-1) +24*(n-2)*(158*n^2-953*n+1445)*a(n-2) +16*(-1232*n^3+13056*n^2-45949*n+53730)*a(n-3) -128*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Jul 28 2014
a(n) = (-1)^(n+1) * 4^(2*n+1) * Sum_{k>=0} (-1/3)^(k+1) * binomial(k/4,n). - Seiichi Manyama, Aug 04 2024

A048966 A convolution triangle of numbers obtained from A025748.

Original entry on oeis.org

1, 3, 1, 15, 6, 1, 90, 39, 9, 1, 594, 270, 72, 12, 1, 4158, 1953, 567, 114, 15, 1, 30294, 14580, 4482, 1008, 165, 18, 1, 227205, 111456, 35721, 8667, 1620, 225, 21, 1, 1741905, 867834, 287199, 73656, 15075, 2430, 294, 24, 1, 13586859, 6857136, 2328183, 623106, 136323, 24354, 3465, 372, 27, 1

Offset: 1

Wolfdieter Lang

A generalization of the Catalan triangle A033184.

			Triangle begins:
     1;
     3,    1;
    15,    6,    1;
    90,   39,    9,    1;
   594,  270,   72,   12,    1;
  4158, 1953,  567,  114,   15,    1;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A034000, A049213, A049223, A049224. a(n, 1)= A025748(n), a(n, 1)= 3^(n-1)*2*A034000(n-1)/n!, n >= 2. Row sums = A025756.

a048966 n k = a048966_tabl !! (n-1) !! (k-1)
a048966_row n = a048966_tabl !! (n-1)
a048966_tabl = [1] : f 2 [1] where
   f x xs = ys : f (x + 1) ys where
     ys = map (flip div x) $ zipWith (+)
          (map (* 3) $ zipWith (*) (map (3 * (x - 1) -) [1..]) (xs ++ [0]))
          (zipWith (*) [1..] ([0] ++ xs))
-- Reinhard Zumkeller, Feb 19 2014

a[n_, m_] /; n >= m >= 1 := a[n, m] = 3*(3*(n-1) - m)*a[n-1, m]/n + m*a[n-1, m-1]/n; a[n_, m_] /; n < m := 0; a[n_, 0] = 0; a[1, 1] = 1; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Apr 26 2011, after given formula *)

a(n, m) = 3*(3*(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-(1-9*x)^(1/3))/3)^m.

a(n,m) = m/n * sum(k=0..n-m, binomial(k,n-m-k) * 3^k*(-1)^(n-m-k) * binomial(n+k-1,n-1)). - Vladimir Kruchinin, Feb 08 2011

A049223 A convolution triangle of numbers obtained from A025750.

Original entry on oeis.org

1, 10, 1, 150, 20, 1, 2625, 400, 30, 1, 49875, 8250, 750, 40, 1, 997500, 174750, 17875, 1200, 50, 1, 20662500, 3780000, 419625, 32500, 1750, 60, 1, 439078125, 83128125, 9810000, 839500, 53125, 2400, 70, 1, 9513359375, 1852500000, 229359375

Offset: 1

Wolfdieter Lang

a(n,1) = A025750(n); a(n,1)= 5^(n-1)*4*A034301(n-1)/n!, n >= 2. G.f. for m-th column: ((1-(1-25*x)^(1/5))/5)^m.

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

Cf. A048966, A049213. Row sums = A025758.

T(n,m):=(m*sum((-1)^(n-m-3*k)*binomial(n+k-1,n-1)*sum(2^j*binomial(k,j)*sum(binomial(j,i-j)*binomial(k-j,n-m-3*(k-j)-i)*5^(3*(k-j)+i),i,j,n-m-k+j),j,0,k),k,0,n-m))/n; /* Vladimir Kruchinin, Dec 10 2011 */

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

T(n,m) = (m*sum(k=0..n-m, (-1)^(n-m-3*k)*binomial(n+k-1,n-1)*sum(j=0..k, 2^j*binomial(k,j)*sum(i=j..n-m-k+j, binomial(j,i-j)*binomial(k-j,n-m-3*(k-j)-i)*5^(3*(k-j)+i)))))/n. - Vladimir Kruchinin, Dec 10 2011

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A000369 Triangle of numbers related to triangle A049213; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.

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A025757 4th-order Vatalan numbers (generalization of Catalan numbers).

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A048966 A convolution triangle of numbers obtained from A025748.

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A049223 A convolution triangle of numbers obtained from A025750.

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