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
-
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
-
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.
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
Comments
a(n,m) := S2(7; n,m) is the seventh triangle of numbers in the sequence S2(k;n,m), k=1..6: A008277 (unsigned Stirling 2nd kind), A008297 (unsigned Lah), A035342, A035469, A049029, A049385, respectively. a(n,1)=A008542(n), n>=1.
a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing 7-ary trees. Proof based on the a(n,m) recurrence. 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
Also the Bell transform of A008542(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016
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
-
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
-
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!.
A223511 Triangle T(n,k) represents the coefficients of (x^9*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.
1, 9, 1, 153, 27, 1, 3825, 855, 54, 1, 126225, 32895, 2745, 90, 1, 5175225, 1507815, 150930, 6705, 135, 1, 253586025, 80565975, 9205245, 499590, 13860, 189, 1, 14454403425, 4926412575, 623675430, 39180645, 1345050, 25578, 252, 1
Offset: 1
Comments
Also the Bell transform of A045755(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016
Examples
1; 9,1; 153,27,1; 3825,855,54,1; 126225,32895,2745,90,1; 5175225,1507815,150930,6705,135,1; 253586025,80565975,9205245,499590,13860,189,1; 14454403425,4926412575,623675430,39180645,1345050,25578,252,1;
Crossrefs
Programs
-
Maple
b[0]:=g(x): for j from 1 to 10 do b[j]:=simplify(x^9*diff(b[j-1],x$1); end do; # The function BellMatrix is defined in A264428. # Adds (1,0,0,0, ..) as column 0. BellMatrix(n -> mul(8*k+1, k=0..n), 10); # Peter Luschny, Jan 29 2016
-
Mathematica
rows = 8; t = Table[Product[8k+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 *)
A223522 Triangle T(n,k) represents the coefficients of (x^20*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.
1, 20, 1, 780, 60, 1, 45240, 4320, 120, 1, 3483480, 382200, 13800, 200, 1, 334414080, 40556880, 1734600, 33600, 300, 1, 38457619200, 5039012160, 243505080, 5699400, 69300, 420, 1
Offset: 1
Examples
1; 20,1; 780,60,1; 45240,4320,120,1; 3483480,382200,13800,200,1; 334414080,40556880,1734600,33600,300,1; 38457619200,5039012160,243505080,5699400,69300,420,1; 5153320972800,718724260800,38155703040,1024322880,15262800,127680,560,1;
Crossrefs
Programs
-
Maple
b[0]:=f(x): for j from 1 to 10 do b[j]:=simplify(x^20*diff(b[j-1],x$1); end do;
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
Comments
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
Examples
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
Links
- 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
Crossrefs
Programs
-
Mathematica
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 *)
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Sage
# 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
Formula
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)
A039810 Matrix square of Stirling2 triangle A008277: 2-levels set partitions of [n] into k first-level subsets.
1, 2, 1, 5, 6, 1, 15, 32, 12, 1, 52, 175, 110, 20, 1, 203, 1012, 945, 280, 30, 1, 877, 6230, 8092, 3465, 595, 42, 1, 4140, 40819, 70756, 40992, 10010, 1120, 56, 1, 21147, 283944, 638423, 479976, 156072, 24570, 1932, 72, 1, 115975, 2090424, 5971350, 5660615, 2350950, 487704, 53550, 3120, 90, 1
Offset: 1
Comments
This triangle groups certain generalized Stirling numbers of the second kind A000558, A000559, ... They can also be interpreted in terms of trees of height 3 with n leaves and constraints on the order of the root.
From Peter Bala, Jul 19 2014: (Start)
The (n,k)-th entry in this table gives the number of double partitions of the set [n] = {1,2,...,n} into k blocks. To form a double partition of [n] we first write [n] as a disjoint union X_1 U...U X_k of k nonempty subsets (blocks) X_i of [n]. Then each block X_i is further partitioned into sub-blocks to give a double partition. For instance, {1,2,4} U {3,5} is a partition of [5] into 2 blocks and {{1,4},{2}} U {{3},{5}} is a refinement of this partition to a double partition of [5] into 2 blocks (and 4 sub-blocks).
Compare the above interpretation for the (n,k)-th entry of this table with the interpretation of the (n,k)-th entry of A013609 (the square of Pascal's triangle but with the rows read in reverse order) as counting the pairs (X,Y) of subsets of [n] such that |Y| = k and X is contained in Y. (End)
Also the Bell transform of the shifted Bell numbers B(n+1) without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016
T(n,k) is the number of partitions of an n-set into colored blocks, such that exactly k colors are used and the colors are introduced in increasing order. T(3,2) = 6: 1a|23b, 13a|2b, 12a|3b, 1a|2a|3b, 1a|2b|3a, 1a|2b|3b. - Alois P. Heinz, Aug 27 2019
Examples
Triangle begins:
k = 1 2 3 4 5 sum
n
1 1 1
2 2 1 3
3 5 6 1 12
4 15 32 12 1 60
5 52 175 110 20 1 358
Matrix multiplication Stirling2 * Stirling2:
1 0 0 0
1 1 0 0
1 3 1 0
1 7 6 1
.
1 0 0 0 1 0 0 0
1 1 0 0 2 1 0 0
1 3 1 0 5 6 1 0
1 7 6 1 15 32 12 1
From _Peter Bala_, Jul 19 2014: (Start)
T(5,2) = 175: A 5-set can be partitioned into 2 blocks as either a union of a 3-set and a 2-set or as a union of a 4-set and a singleton set.
In the first case there are 10 ways of partitioning a 5-set into a 3-set and a 2-set. Each 3-set can be further partitioned into sub-blocks in Bell(3) = 5 ways and each 2-set can be further partitioned into sub-blocks in Bell(2) = 2 ways. So altogether we obtain 10*5*2 = 100 double partitions of this type.
In the second case, there are 5 ways of partitioning a 5-set into a 4-set and a 1-set. Each 4-set can be further partitioned in Bell(4) = 15 ways and each 1-set can be further partitioned in Bell(1) = 1 way. So altogether we obtain 5*15*1 = 75 double partitions of this type.
Hence, in total, T(5,2) = 100 + 75 = 175. (End)
Links
- Tilman Piesk, First 100 rows, flattened
- A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, and O. Mallet, Bell polynomials in combinatorial Hopf algebras, arXiv preprint arXiv:1402.2960 [math.CO], 2014-2015.
- T. Mansour, A. Munagi, and M. Shattuck, Recurrence Relations and Two-Dimensional Set Partitions , J. Int. Seq. 14 (2011) # 11.4.1.
Crossrefs
T(n, 1) = A000110(n) (first column) (Bell numbers).
T(n, 2) = A000558(n) 2-levels set partitions with 2 first-level classes.
T(n, n-1) = A002378(n-1) = n*(n-1) = 2*C(n,2) = set-partitions into (n-2) singletons and one of the two possible set partitions of [2].
Sum is A000258(n), 2-levels set partitions.
Another version with offset 0: A130191.
Horizontal mirror triangle is A046817.
T(2n,n) gives A321712.
Programs
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Maple
# The function BellMatrix is defined in A264428. # Adds (1,0,0,0, ..) as column 0. BellMatrix(n -> combinat:-bell(n+1), 10); # Peter Luschny, Jan 28 2016
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Mathematica
Flatten[Table[Sum[StirlingS2[n,i]*StirlingS2[i,k],{i,k,n}],{n,1,10},{k,1,n}]] (* Indranil Ghosh, Feb 22 2017 *) rows = 10; t = Table[BellB[n+1], {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 *) -
PARI
T(n, k) = sum(j=0, n, stirling(n, j, 2)*stirling(j, k, 2)); \\ Seiichi Manyama, Feb 13 2022
Formula
S2 = A008277 (Stirling numbers of the second kind).
T = (S2)^2.
T(n,k) = Sum_{i=k..n} S2(n,i) * S2(i,k).
E.g.f. of k-th column: (exp(exp(x)-1)-1)^k/k!. [corrected by Seiichi Manyama, Feb 12 2022]
From Peter Bala, Jul 19 2014: (Start)
T(n,k) = Sum_{disjoint unions X_1 U...U X_k = [n]} Bell(|X_1|)*...*Bell(|X_k|), where Bell(n) = A000110(n).
Recurrence equation: T(n+1,k+1) = Sum_{j = k..n} Bell(n+1-j)*binomial(n,j)* T(j,k).
Row sums [1,3,12,60,358,...] = A000258. (End)
Extensions
Definition and interpretation edited by Olivier Gérard, Jul 31 2011
A102661 Triangle of partial sums of Stirling numbers of 2nd kind (A008277): T(n,k) = Sum_{i=1..k} Stirling2(n,i), 1<=k<=n.
1, 1, 2, 1, 4, 5, 1, 8, 14, 15, 1, 16, 41, 51, 52, 1, 32, 122, 187, 202, 203, 1, 64, 365, 715, 855, 876, 877, 1, 128, 1094, 2795, 3845, 4111, 4139, 4140, 1, 256, 3281, 11051, 18002, 20648, 21110, 21146, 21147, 1, 512, 9842, 43947, 86472, 109299, 115179, 115929, 115974, 115975
Offset: 1
Comments
T(n,k) is the number of ways to place n distinguishable balls into k indistinguishable bins. - Geoffrey Critzer, Mar 22 2011
From Mark Wildon, Aug 10 2015: (Start)
T(n,k) is the number of partitions of a set of size n into at most k parts.
T(n,k) is the number of sequences of n top-to-random shuffles of a deck of k cards that leave the deck invariant.
T(n,k) = where pi is the natural permutation character of the symmetric group Sym_k. This gives another combinatorial interpretation of T(n,k) as counting sequences of box moves on Young diagrams. Reference linked to below. (End)
Diagonal entries T(n,n) are the Bell numbers A000110. - Robert Israel, Aug 10 2015
From Manfred Boergens, Mar 18 2025: (Start)
The partitions in the second comment can be described as disjoint collections of subsets of [n] without the empty set with union = [n]. For instance, T(4,2)=8 is the number of partitions of [4] into 1 or 2 parts: 1234, 1 234, 2 134, 3 124, 4 123, 12 34, 13 24, 14 23.
For disjoint collections which may include one empty set see A381682.
For arbitrary collections without the empty set see A369950.
For arbitrary collections which may include one empty set see A381683. (End)
Examples
Triangle begins: 1; 1, 2; 1, 4, 5; 1, 8, 14, 15; 1, 16, 41, 51, 52; ...
References
- Richard Stanley, Enumerative Combinatorics, Cambridge Univ. Press, 1997 page 38. (#7 of the twelvefold ways)
Links
- Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
- John R. Britnell and Mark Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D, arXiv:1507.04803 [math.CO], 2015.
- T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
- OEIS Wiki, Sorting numbers
Programs
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Haskell
a102661 n k = a102661_tabl !! (n-1) !! (k-1) a102661_row n = a102661_tabl !! (n-1) a102661_tabl = map (scanl1 (+) . tail) $ tail a048993_tabl -- Reinhard Zumkeller, Jun 19 2015
-
Maple
with(combinat): A102661_row := proc(n) local k,j; seq(add(stirling2(n,j),j=1..k),k=1..n) end: seq(print(A102661_row(r)),r=1..6); # Peter Luschny, Sep 30 2011
-
Mathematica
Table[Table[Sum[StirlingS2[n, i], {i, 1, k}], {k, 1, n}], {n, 1,10}] // Grid (* Geoffrey Critzer, Mar 22 2011*) Table[Accumulate[StirlingS2[n,Range[n]]],{n,10}]//Flatten (* Harvey P. Dale, Oct 28 2019 *) -
PARI
tabl(nn) = {for (n=1, nn, for (k=1, n, print1(sum(i=1, k, stirling(n,i, 2)), ", ");); print(););} \\ Michel Marcus, Aug 10 2015 -
Sage
def T(n,k): return sum([stirling_number2(n,j) for j in range(1,k+1)]) # Danny Rorabaugh, Oct 13 2015
Formula
E.g.f. for row polynomials s(n,y) = Sum_{k=0..n} a(n,k)*y^k is (y*e^(e^(x*y)-1)- e^(y*(e^x-1)))/(y-1) - 1. - Robert Israel, Aug 10 2015
A039813 Matrix 5th power of Stirling2 triangle A008277.
1, 5, 1, 35, 15, 1, 315, 215, 30, 1, 3455, 3325, 725, 50, 1, 44590, 56605, 17100, 1825, 75, 1, 660665, 1060780, 415555, 60900, 3850, 105, 1, 11035095, 21772595, 10606470, 1998605, 172550, 7210, 140, 1, 204904830, 486459105, 286281665, 66528210, 7346955, 417690, 12390, 180, 1
Offset: 1
Examples
Triangle begins:
1;
5, 1;
35, 15, 1;
315, 215, 30, 1;
3455, 3325, 725, 50, 1;
44590, 56605, 17100, 1825, 75, 1;
...
Links
- Seiichi Manyama, Rows n = 1..140, flattened
Programs
-
Mathematica
max = 9; m = MatrixPower[Array[StirlingS2, {max, max}], 5]; Table[Take[m[[n]], n], {n, 1, max}] // Flatten (* Jean-François Alcover, Mar 03 2014 *)
Formula
E.g.f. k-th column: (( exp(exp(exp(exp(exp(x)-1)-1)-1)-1)-1 )^k)/k!. [corrected by Seiichi Manyama, Feb 12 2022]
A039811 Triangle read by rows: matrix cube of the Stirling2 triangle A008277.
1, 3, 1, 12, 9, 1, 60, 75, 18, 1, 358, 660, 255, 30, 1, 2471, 6288, 3465, 645, 45, 1, 19302, 65051, 47838, 12495, 1365, 63, 1, 167894, 728556, 685580, 235193, 35700, 2562, 84, 1, 1606137, 8792910, 10285488, 4444188, 877653, 86940, 4410, 108, 1
Offset: 1
Examples
Triangle begins
1;
3, 1;
12, 9, 1;
60, 75, 18, 1;
358, 660, 255, 30, 1;
2471, 6288, 3465, 645, 45, 1;
...
Links
- Seiichi Manyama, Rows n = 1..140, flattened
Programs
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Mathematica
Flatten[Table[SeriesCoefficient[(Exp[Exp[Exp[x]-1]-1]-1)^k, {x,0,n}] n!/k!,{n,9}, {k,n}]] (* Stefano Spezia, Sep 12 2022 *)
Formula
E.g.f. k-th column: (( exp(exp(exp(x)-1)-1)-1 )^k)/k!. [corrected by Seiichi Manyama, Feb 12 2022]
A079641 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Stirling1-triangle |A008275(n,k)|.
1, 2, 1, 6, 6, 1, 26, 36, 12, 1, 150, 250, 120, 20, 1, 1082, 2040, 1230, 300, 30, 1, 9366, 19334, 13650, 4270, 630, 42, 1, 94586, 209580, 166376, 62160, 11900, 1176, 56, 1, 1091670, 2562354, 2229444, 952728, 220500, 28476, 2016, 72, 1, 14174522
Offset: 1
Comments
Triangle T(n,k), 1<=k<=n, read by rows, given by (0, 2, 1, 4, 2, 6, 3, 8, 4, 10, 5, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 22 2011
Subtriangle of triangle in A129062. - Philippe Deléham, Feb 17 2013
Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016
Examples
Triangle begins: 1; 2,1; 6,6,1; 26,36,12,1; 150,250,120,20,1; 1082,2040,1230,300,30,1; ... Triangle (0,2,1,4,2,6,3,8,4,...) DELTA (1,0,1,0,1,0,1,0,1,...) begins: 1 0, 1 0, 2, 1 0, 6, 6, 1 0, 26, 36, 12, 1 0, 150, 250, 120, 20, 1 0, 1082, 2040, 1230, 300, 30, 1. - _Philippe Deléham_, Dec 22 2011
Links
- Nick Early, Canonical Bases for Permutohedral Plates, arXiv:1712.08520 [math.CO], 2017.
- Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
- D. E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA], 1992; The Mathematica J., 2 (1992), 67-78.
Programs
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Maple
# The function BellMatrix is defined in A264428. # Adds (1, 0, 0, 0, ..) as column 0. BellMatrix(n -> add((-1)^(n-k)*2^k*k!*combinat:-stirling2(n, k), k=0..n), 9); # Peter Luschny, Jan 26 2016
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Mathematica
rows = 10; t = Table[Sum[(-1)^(n-k)*2^k*k!*StirlingS2[n, k], {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 *)
Formula
E.g.f.: (2-exp(x))^(-y). - Vladeta Jovovic, Nov 22 2003
From Peter Bala, Sep 12 2011: (Start)
The row generating polynomials R(n,x) begin R(1,x) = x, R(2,x) = 2*x + x^2, R(3,x) = 6*x + 6*x^2 + x^3 and satisfy the recurrence R(n+1,x) = x*(2*R(n,x+1) - R(n,x)). They form a sequence of binomial type polynomials. In particular, denoting R(n,x) by x^[n] to emphasize the analogies with the monomial polynomials x^n, we have the binomial expansion (x + y)^[n] = Sum_{k = 0..n} binomial(n,k)*x^[n-k]*y^[k].
There is a Dobinski-type formula: exp(-x)*Sum_{k >= 0} (-k)^[n] * x^k/k! = Bell(n,-x). The alternating n-th row entries (-1)^k * T(n,k) are the connection coefficients expressing the polynomial Bell(n,-x) as a linear combination of Bell(k,x), 1 <= k <= n. For example, the list of coefficients of R(4,x) is [26, 36, 12, 1] and we have Bell(4,-x) = -26*Bell(1,x) + 36*Bell(2,x) - 12*Bell(3,x) + Bell(4,x).
The row polynomials also satisfy an analog of the Bernoulli's summation formula for powers of integers, namely, Sum_{k = 1..n} k^[p] = 1/(p+1) * Sum_{k = 0..p} binomial(p+1,k) * B_k * n^[p+1-k], where B_k denotes the Bernoulli numbers. Compare with A195204 and A195205. (End)
Let D be the forward difference operator D(f(x)) = f(x+1) - f(x). Then the n-th row polynomial R(n,x) = 1/f(x) * (x*D)^n(f(x)) with f(x) = 2^x. Cf. A209849. Also cf. A008277, where the row polynomials are given by 1/f(x) * (x*d/dx)^n(f(x)), where now f(x) = exp(x). - Peter Bala, Mar 16 2012
Conjecture: o.g.f. as a continued fraction of Stieltjes type: 1/(1 - x*z/(1 - 2*z/(1 - (x + 1)*z/(1 - 4*z/(1 - (x + 2)*z/(1 - 6*z/(1 - (x + 3)*z/(1 - 8*z/(1 - ... ))))))))) = 1 + x*z + (2*x + x^2)*z^2 + (6*x + 6*x^2 + x^3)*z^3 + .... - Peter Bala, Dec 12 2024
Comments