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 *)
A002868 Largest number in n-th row of triangle of Lah numbers (A008297 and A271703).
1, 1, 2, 6, 36, 240, 1800, 15120, 141120, 1693440, 21772800, 299376000, 4390848000, 68497228800, 1133317785600, 19833061248000, 396661224960000, 8299373322240000, 181400588328960000, 4135933413900288000, 98228418580131840000, 2426819753156198400000
Offset: 0
References
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..100
- Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
- 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
-
Haskell
a002868 n = if n == 0 then 1 else maximum $ map abs $ a008297_row n -- Reinhard Zumkeller, Sep 30 2014
-
Maple
with(combinat): for n from 0 to 35 do big := 1: for m from 1 to n do if big < n!*binomial(n-1,m-1)/m! then big := n!*binomial(n-1,m-1)/m! fi: od: printf(`%d,`,big): od:
-
Mathematica
a[n_] := ( big = 1; For[ m = 1 , m <= n, m++, b = n!*Binomial[n - 1, m - 1]/m!; If[ big < b , big = b ]]; big); Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Sep 21 2012, after Maple *)
Formula
For 2 <= n <= 7, equals (n+1)!*n/2. - Alexander R. Povolotsky, Oct 16 2006
Extensions
More terms from James Sellers, Jan 03 2001
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;
A187535 Central Lah numbers: a(n) = A105278(2*n,n) = A008297(2*n,n).
1, 2, 36, 1200, 58800, 3810240, 307359360, 29682132480, 3339239904000, 428906814336000, 61934143990118400, 9931984545324441600, 1751339941492209868800, 336796142594655744000000, 70149825129001153536000000, 15732267448930658699673600000
Offset: 0
Comments
a(n) is the number of Lah partitions of a set of size 2n with n blocks.
Crossrefs
Programs
-
Maple
A187535:= n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi; seq(A187535(n),n=0..12);
-
Mathematica
a[n_]:=If[n==0,1,Binomial[2n-1,n-1](2n)!/n!] Table[a[n],{n,0,12}] (* Alternative: *) a[n_] := Binomial[2*n, n] FactorialPower[2*n - 1, n]; Table[a[n], {n, 0, 15}] (* Peter Luschny, Jun 15 2022 *) -
Maxima
a(n) := if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!; makelist(a(n),n,0,12);
-
Sage
[catalan_number(n)*binomial(2*n-1,n)*factorial(n+1) for n in range(15)] # Peter Luschny, Oct 07 2014
Formula
a(n) = binomial(2n-1,n-1)*(2n)!/n! (for n>0).
D-finite with recurrence (n+1)*a(n+1) = 4*(2n+1)^2*a(n) - delta(n,0).
a(n) ~ 2^(4*n)*n^n*exp(-n)/sqrt(2*n*Pi).
a(n)*a(n+2) - a(n+1)^2 is >= 0 and is a multiple of 2^(n+3) for all nonnegative n.
a(n) == 0 (mod 10) for n>3.
E.g.f.: 1/2 + K(16x)/Pi, where K(z) is the complete elliptic integral of the first kind, which can also be written as a Legendre function of the second kind.
a(n) = Catalan(n)*C(2*n-1,n)*(n+1)!. - Peter Luschny, Oct 07 2014
a(n) = (2/n)*(Gamma(2*n)^2/Gamma(n)^3) for n>0. - Peter Luschny, Oct 17 2014
A089231 Triangular array A066667 or A008297 unsigned and transposed.
1, 1, 2, 1, 6, 6, 1, 12, 36, 24, 1, 20, 120, 240, 120, 1, 30, 300, 1200, 1800, 720, 1, 42, 630, 4200, 12600, 15120, 5040, 1, 56, 1176, 11760, 58800, 141120, 141120, 40320, 1, 72, 2016, 28224, 211680, 846720, 1693440, 1451520, 362880
Offset: 1
Comments
Row sums: A000262.
T(n, k) is also the number of nilpotent partial one-one bijections (of an n-element set) of height k (height(alpha) = |Im(alpha)|). - Abdullahi Umar, Sep 14 2008
T(n, k) is also the number of acyclic directed graphs on n labeled nodes with k-1 edges with all indegrees and outdegrees at most 1. - Felix A. Pahl, Dec 25 2012
For n > 1, the n-th derivative of exp(1/x) is of the form (exp(1/x)/x^(2*n))*(P(n-1,x)) where P(n-1,x) is a polynomial of degree n-1 with n terms. The term of degree k in P(n-1,x) has a coefficient given by T(n-1,k). Example: The third derivative of exp(1/x) is (exp(1/x)/x^6)*(1+6x+6x^2) and the 3rd row of this triangle is 1, 6, 6, which corresponds to this coefficients of the polynomial 1+6x+6x^2. - Derek Orr, Nov 06 2014
For another context for this array see the Callan (2008) article. - Ron L.J. van den Burg, Dec 12 2021
Examples
1; 1, 2; 1, 6, 6; 1, 12, 36, 24; 1, 20, 120, 240, 120; 1, 30, 300, 1200, 1800, 720; 1, 42, 630, 4200, 12600, 15120, 5040; 1, 56, 1176, 11760, 58800, 141120, 141120, 40320; 1, 72, 2016, 28224, 211680, 846720, 1693440, 1451520, 362880;
References
- A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 203.
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
- David Callan, Sets, Lists and Noncrossing Partitions, Journal of Integer Sequences, Vol. 11 (2008), Article 08.1.3. Also on arXiv, arXiv:0711.4841 [math.CO], 2007-2008.
- Tom Copeland, Lagrange a la Lah, 2011.
- Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015
- Olexandr Ganyushkin and Volodymyr Mazorchuk, Combinatorics of nilpotents in symmetric inverse semigroups, Ann. Comb. 8 (2004), no. 2, 161--175. [From _Abdullahi Umar_, Sep 14 2008]
- F. Hivert, J.-C. Novelli and J.-Y. Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006.
- Matthieu Josuat-Vergès, Stammering tableaux - Tableaux bégayants, arXiv:1601.02212 [math.CO], 2016. See Lemma 7.1 p. 16.
- A. Laradji and A. Umar, On the number of nilpotents in the partial symmetric semigroup, Comm. Algebra 32 (2004), 3017-3023.
- Jair Taylor, Number of acyclic digraphs on [n] with k edges and each indegree, outdegree <=1 (question on StackExchange)
- Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.
Programs
-
Maple
P := n -> simplify(hypergeom([-n,-n+1],[],1/t)); seq(print(seq(coeff(expand(t^k*P(k)),t,k-j+1),j=1..k)),k=1..n); # Peter Luschny, Oct 29 2014
-
Mathematica
Table[(Binomial[n - 1, k - 1] Binomial[n, k - 1]/k) k!, {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jul 04 2016 *) -
PARI
tabl(nn) = {for (n=0, nn, for (k=0, n, print1((n+1)!*binomial(n,k)/(n-k+1)!, ", ");); print(););} \\ Michel Marcus, Jan 12 2016
Formula
T(n, k) = C(n, n-k+1)*(n-1)!/(n-k)! = Sum_{i=n-k+1..n} |S1(n, i)*S2(i, n-k+1)| , with S1, S2 the Stirling numbers.
From Derek Orr, Mar 12 2015: (Start)
Each row represents a polynomial:
P(1,x) = 1;
P(2,x) = 1 + 2x;
P(3,x) = 1 + 6x + 6x^2;
P(4,x) = 1 + 12x + 36x^2 + 24x^3;
...
They are related through P(n+1,x) = x^2*P'(n,x) - (1+2*n*x)*P(n,x) with P(1,x) = 1.
(End)
From Peter Bala, Jul 04 2016: (Start)
Working with an offset of 0:
G.f.: exp(x*t)*I_1(2*sqrt(x)) = 1 + (1 + 2*t)*x/(1!*2!) + (1 + 6*t + 6*t^2)*x^2/(2!*3!) + (1 + 12*t + 36*t^2 + 24*t^3)*x^3/(3!*4!) + ..., where I_1(x) = Sum_{n >= 0} (x/2)^(2*n)/(n!*(n+1)!) is a modified Bessel function of the first kind.
The row polynomials R(n,t) satisfy R(n,t + u) = Sum_{k = 0..n} T(n,k)*t^k*R(n-k,u).
R(n,t) = 1 + Sum_{k = 0..n-1} (-1)^(n-k+1)*(n+1)!/(k+1)!* binomial(n,k)*t^(n-k)*R(k,t). Cf. A144084. (End)
From Peter Bala, Oct 05 2019: (Start)
The following formulas use a column index k starting at 0:
E.g.f.: exp(x/(1 - t*x)) - 1 = x + (1 + 2*t)*x^2/2! + (1 + 6*t + 6*t^2)*x^3/3! + ....
Recurrence for row polynomials: R(n+1,t) = (1 + 2*n*t)R(n,t) - n*(n-1)*t^2*R(n-1,t), with R(1,t) = 1 and R(2,t) = 1 + 2*t.
R(n+1,t) equals the numerator polynomial of the finite continued fraction 1 + n*t/(1 + n*t/(1 + (n-1)*t/(1 + (n-1)*t/(1 + ... + 2*t/(1 + 2*t/(1 + t/(1 + t/(1)))))))). The denominator polynomial is the n-th row polynomial of A144084. (End)
T(n,k) = A105278(n,n-k). - Ron L.J. van den Burg, Dec 12 2021
Extensions
StackExchange link added by Felix A. Pahl, Dec 25 2012
A136657 Unsigned member s=2 of a family of generalizations of the (signed) Lah triangle A008297. All numbers divided by 2.
1, 3, 2, 12, 18, 4, 60, 150, 72, 8, 360, 1320, 1020, 240, 16, 2520, 12600, 13860, 5160, 720, 32, 20160, 131040, 191520, 99960, 21840, 2016, 64, 181440, 1481760, 2751840, 1882440, 571200, 81984, 5376, 128, 1814400, 18144000, 41489280, 35622720
Offset: 0
Comments
In order to obtain the Lah triangle for s=+1 the sign of the s parameter in the Charalambides reference has been switched.
For more information see entry A136656 and the Charalambides reference.
Examples
[1];[3,2];[12,18,4];[60,150,72,8];[360,1320,1020,240,16];...
References
- Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, ch. 8.4 p. 301 ff, with s -> -s. Table 8.3 for s=-2 and multiplied by(-1)^n, divided by 2.
Links
- W. Lang, First ten rows and more.
Programs
-
Mathematica
fallfac[n_, k_] := Pochhammer[n - k + 1, k]; a[n_, k_] := Sum[(-1)^(k - r)*Binomial[k, r]*fallfac[-2*r, n], {r, 0, k}]/(2*k!); Table[(-1)^n*a[n, k], {n, 0, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 09 2013 *)
Formula
a(n,k)=sum(((-1)^(k-r))*binomial(k,r)*fallfac(-2*r,n),r=0..k)/(2*k!), n>=k>=1. From the Charalambides reference Theorem 8.15, p. 306 for s=-2, divided by 2.
A079640 Matrix product of unsigned Stirling1-triangle |A008275(n,k)| and unsigned Lah-triangle |A008297(n,k)|.
1, 3, 1, 14, 9, 1, 88, 83, 18, 1, 694, 860, 275, 30, 1, 6578, 10084, 4245, 685, 45, 1, 72792, 132888, 69244, 14735, 1435, 63, 1, 920904, 1950024, 1209880, 318969, 41020, 2674, 84, 1, 13109088, 31580472, 22715972, 7133784, 1137549, 98028, 4578, 108, 1
Offset: 1
Comments
Also the Bell transform of A007840(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016
Examples
1; 3,1; 14,9,1; 88,83,18,1; 694,860,275,30,1; 6578,10084,4245,685,45,1; ...
Crossrefs
Cf. A007840 (first column).
Programs
-
Maple
# The function BellMatrix is defined in A264428. # Adds (1, 0, 0, 0, ..) as column 0. BellMatrix(n -> add(k!*abs(combinat:-stirling1(n+1, k)), k=0..n+1), 9); # Peter Luschny, Jan 26 2016
-
Mathematica
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]]; rows = 12; M = BellMatrix[Function[n, Sum[k!*Abs[StirlingS1[n+1, k]], {k, 0, n+1}]], rows]; Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 26 2018, after Peter Luschny *)
Formula
E.g.f.: (1-x)^(-y/(1+log(1-x))). - Vladeta Jovovic, Nov 22 2003
A079638 Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.
1, 3, 1, 14, 9, 1, 90, 83, 18, 1, 744, 870, 275, 30, 1, 7560, 10474, 4275, 685, 45, 1, 91440, 143892, 70924, 14805, 1435, 63, 1, 1285200, 2233356, 1274196, 324289, 41160, 2674, 84, 1, 20603520, 38769840, 24870572, 7398972, 1151409, 98280, 4578
Offset: 1
Comments
Matrix product of unsigned Lah-triangle |A008297(n,k)| and Stirling1-triangle A008275(n,k) is unsigned Stirling1-triangle |A008275(n,k)|.
Also the Bell transform of A029767(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016
Essentially the same as A131222. - Peter Bala, Feb 12 2022
Examples
Triangle begins
1;
3, 1;
14, 9, 1;
90, 83, 18, 1;
744, 870, 275, 30, 1;
7560, 10474, 4275, 685, 45, 1;
...
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..1225 (rows n = 1..50, flattened).
- William Keith, Rishi Nath, and James Sellers, On simultaneous (s, s+t, s+2t, ...)-core partitions, arXiv:2508.00074 [math.CO], 2025. See p. 3.
Programs
-
Maple
# The function BellMatrix is defined in A264428. # Adds (1, 0, 0, 0, ..) as column 0. BellMatrix(n -> n!*(2^(n+1)-1), 9); # Peter Luschny, Jan 26 2016
-
Mathematica
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]]; B = BellMatrix[Function[n, n! (2^(n + 1) - 1)], rows = 12]; Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)
Formula
E.g.f.: ((1-x)/(1-2*x))^y. - Vladeta Jovovic, Nov 22 2003
Comments