A016036 -id:A016036 - 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)

A157403 A partition product of Stirling_2 type [parameter k = 3] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 3, 1, 9, 21, 1, 45, 84, 231, 1, 165, 840, 1155, 3465, 1, 855, 8610, 13860, 20790, 65835, 1, 3843, 64680, 250635, 291060, 460845, 1514205, 1, 21819, 689136, 3969735, 6015240, 7373520, 12113640, 40883535, 1, 114075

Offset: 1

Peter Luschny, Mar 09 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 3,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A143173.
Same partition product with length statistic is A000369.
Diagonal a(A000217) = A008545
Row sum is A016036.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157404, A157405

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(4*j - 1).

A028575 Row sums of triangle A011801.

Original entry on oeis.org

1, 5, 49, 721, 14177, 349141, 10334689, 357361985, 14137664833, 629779342213, 31195027543505, 1700812505769169, 101218448336028193, 6528869281965115541, 453720852957751220353, 33796334125623555379969, 2686138908337714715560577, 226908450494953996837748869

Offset: 1

Wolfdieter Lang

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Sequences with e.g.f. exp(1-(1-m*x)^(1/m)) - 1: A000012 (m=1), A001515 (m=2), A015735 (m=3), A016036 (m=4), this sequence (m=5), A028844 (m=6).

Cf. A011801.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-5*x)^(1/5)) - 1 ))); // G. C. Greubel, Oct 02 2023

Rest[With[{nn=20},CoefficientList[Series[Exp[1-(1-5x)^(1/5)]-1, {x,0,nn}], x] Range[0,nn]!]] (* Harvey P. Dale, Aug 02 2016 *)

def A028575_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(1-(1-5*x)^(1/5)) -1 ).egf_to_ogf().list()
a=A028575_list(40); a[1:] # G. C. Greubel, Oct 02 2023

E.g.f.: exp(1 - (1-5*x)^(1/5)) - 1.
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x)^4*d/dx. Cf. A001515, A015735 and A016036. - Peter Bala, Nov 25 2011
D-finite with recurrence: a(n) -20*(n-3)*a(n-1) +30*(5*n^2-35*n +62)*a(n-2) -100*(n-4)*(5*n^2-40*n+81)*a(n-3) +(5*n-22)*(5*n-21)*(5*n-24)*(5*n-23)*a(n-4) -a(n-5) = 0. - R. J. Mathar, Jan 28 2020
From Seiichi Manyama, Jan 20 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * |Stirling1(n,k)| * A000587(k).
a(n) = e * (-5)^n * n! * Sum_{k>=0} (-1)^k * binomial(k/5,n)/k!. (End)

A015735 Row sums of triangle A004747.

Original entry on oeis.org

1, 3, 17, 145, 1661, 23931, 415773, 8460257, 197360985, 5192853011, 152137882601, 4911873672113, 173268075672277, 6630323916472075, 273555262963272501, 12105084133976359361, 571897644855277242673, 28731255563712689630627, 1529450942687399074134465

Offset: 1

Wolfdieter Lang

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A001515, A004747, A016036, A028575.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-3*x)^(1/3)) - 1 ))); // G. C. Greubel, Oct 02 2023

a[1]=1; a[n_]:= 1 +(n-1)!*Sum[Binomial[k, n-m-k]*Binomial[k+n-1,n-1]*(-1/3)^(n-m-k)/(m-1)!, {m,n}, {k,n-m}]; Table[a[n], {n,20}] (* Jean-François Alcover, Jul 05 2013, after Vladimir Kruchinin *)
Rest@With[{m=30}, CoefficientList[Series[Exp[1-Surd[1-3*x,3]] -1, {x, 0,m}], x]*Range[0,m]!] (* G. C. Greubel, Oct 02 2023 *)

a(n):=if n=1 then 1 else (n-1)!*sum(sum(binomial(k,n-m-k)* (-1/3)^(n-m-k)*binomial(k+n-1,n-1),k,1,n-m)/(m-1)!,m,1,n)+1; /* Vladimir Kruchinin, Aug 08 2010 */

def A015735_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(1-(1-3*x)^(1/3)) -1 ).egf_to_ogf().list()
a=A015735_list(40); a[1:] # G. C. Greubel, Oct 02 2023

E.g.f.: exp(1-(1-3*x)^(1/3)) - 1, if one takes a(0)=0.
a(n) = 6*(n-2)*a(n-1) - (3*n-8)*(3*n-7)*a(n-2) + a(n-3), a(0)=1, a(1)=1, a(2)=3.
a(n) = 1 + (n-1)!*Sum_{m=1..n} ( Sum_{k=1..n-m} C(k, n-m-k)*C(k+n-1, n-1)*(-1/3)^(n-m-k) ) / (m-1)!, n > 1. - Vladimir Kruchinin, Aug 08 2010
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x)^2*d/dx. Cf. A001515, A016036 and A028575. - Peter Bala, Nov 25 2011
E.g.f. with offset 0: exp(1-(1-3*x)^(1/3))/(1-3*x)^(2/3). - Sergei N. Gladkovskii, Jul 07 2012.
a(n) ~ sqrt(2*Pi)*3^(n-1)*exp(1-n)*n^(n-5/6)/Gamma(2/3) * (1-sqrt(3)*Gamma(2/3)^2/(2*Pi*n^(1/3))). - Vaclav Kotesovec, Aug 10 2013
From Seiichi Manyama, Jan 20 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 3^(n-k) * |Stirling1(n,k)| * A000587(k).
a(n) = e * (-3)^n * n! * Sum_{k>=0} (-1)^k * binomial(k/3,n)/k!. (End)

A028844 Row sums of triangle A013988.

Original entry on oeis.org

1, 6, 71, 1261, 29906, 887751, 31657851, 1318279586, 62783681421, 3365947782611, 200610405843926, 13157941480889921, 941848076798467801, 73060842413607398806, 6105266987293752470991, 546770299628690541571901, 52244284936267317229542466, 5305131708827069245129523591

Offset: 1

Wolfdieter Lang

nonn

G. C. Greubel, Table of n, a(n) for n = 1..250
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Sequences with e.g.f. exp(1-(1-m*x)^(1/m)) - 1: A000012 (m=1), A001515 (m=2), A015735 (m=3), A016036 (m=4), A028575 (m=5), this sequence (m=6).

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-6*x)^(1/6)) -1 ))); // G. C. Greubel, Oct 03 2023

With[{nn=20},Rest[CoefficientList[Series[Exp[1-(1-6x)^(1/6)]-1,{x,0,nn}], x]Range[0,nn]!]] (* Harvey P. Dale, Feb 02 2012 *)

def A028844_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(1-(1-6*x)^(1/6)) -1 ).egf_to_ogf().list()
a=A028844_list(40); a[1:] # G. C. Greubel, Oct 03 2023

E.g.f.: exp(1 - (1-6*x)^(1/6)) - 1.
D-finite with recurrence: a(n) = 15*(2*n-7)*a(n-1) +5*(72*n^2-576*n+1169)*a(n-2) +45*(2*n-9)*(24*n^2-216*n+497)*a(n-3) -20*(324*n^4-6480*n^3+48735*n^2-163350*n+205877)*a(n-4) +12*(6*n-35)*(6*n-31)*(3*n-16)*(2*n-11)*(3*n-17)*a(n-5) +a(n-6). - R. J. Mathar, Jan 28 2020
From Seiichi Manyama, Jan 20 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * |Stirling1(n,k)| * A000587(k).
a(n) = e * (-6)^n * n! * Sum_{k>=0} (-1)^k * binomial(k/6,n)/k!. (End)

A132063 Alternating row sums of Jabotinsky type triangle S2p(-3):= A000369.

Original entry on oeis.org

1, 2, 13, 137, 1996, 37109, 838993, 22336292, 684256123, 23705447669, 916278718246, 39088637286467, 1824146816097373, 92434987026468722, 5054207258536066681, 296598839923053166109, 18593246768511584995468

Offset: 1

Wolfdieter Lang Sep 14 2007

Cf. A016036 (row sums of A000369).

a(n)=sum(A000369(n,m)*(-1)^(m-1),m=1..n)

E.g.f.: 1-exp(-(1-(1-4*x)^(1/4))).

Showing 1-6 of 6 results.

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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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A157403 A partition product of Stirling_2 type [parameter k = 3] with biggest-part statistic (triangle read by rows).

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A028575 Row sums of triangle A011801.

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A015735 Row sums of triangle A004747.

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A028844 Row sums of triangle A013988.

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A132063 Alternating row sums of Jabotinsky type triangle S2p(-3):= A000369.

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