A049412 -id:A049412 - cp's OEIS frontend

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

Wolfdieter Lang

a(n,m) := S2(6; n,m) is the sixth triangle of numbers in the sequence S2(k; n,m), k=1..6: A008277 (unsigned Stirling 2nd kind), A008297 (unsigned Lah), A035342, A035469, A049029, respectively. a(n,1)= A008548(n).

a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing 6-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

			Triangle begins:
  {1};
  {6,1};
  {66,18,1};
  {1056,372,36,1};
  ...

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.

Cf. A049412.

# 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

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 *)

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.

A134278 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6).

Original entry on oeis.org

1, 6, 1, 66, 18, 1, 1056, 264, 108, 36, 1, 22176, 5280, 3960, 660, 540, 60, 1, 576576, 133056, 95040, 43560, 15840, 23760, 3240, 1320, 1620, 90, 1, 17873856, 4036032, 2794176, 2439360, 465696, 665280, 304920, 249480, 36960, 83160, 22680, 2310

Offset: 1

Wolfdieter Lang, Nov 13 2007

For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_3(6), the k=6 member in the family of a generalization of the multinomial number arrays M_3 = M_3(1) = A036040.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
The S2(6,n,m):=A049385(n,m) numbers (generalized Stirling2 numbers) are obtained by summing in row n all numbers with the same part number m. In the same manner the S2(n,m) (Stirling2) numbers A008277 are obtained from the partition array M_3= A036040.
a(n,k) enumerates unordered forests of increasing 6-ary trees related to the k-th partition of n in the A-St order. The m-forest is composed of m such trees, with m the number of parts of the partition.

			[1]; [6,1]; [66,18,1]; [1056,264,108,36,1]; [22176,5280,3960,660,540,60,1]; ...
There are a(4,3) = 108 = 3*6^2 unordered 2-forests with 4 vertices, composed of two 6-ary increasing trees, each with two vertices: there are 3 increasing labelings (1,2)(3,4); (1,3)(2,4); (1,4)(2,3) and each tree comes in six versions from the 6-ary structure.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Wolfdieter Lang, First 10 rows and more.

Cf. A049412 (row sums, also of triangle A049385).

Cf. A134273 (M_3(5) partition array).

a(n,k) = n!*Product_{j=1..n} (S2(6,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(6,n,1) = A049385(n,1) = A008548(n) = (5*n-4)(!^5) (quintuple- or 5-factorials) and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1.

A157396 A partition product of Stirling_2 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 6, 1, 18, 66, 1, 144, 264, 1056, 1, 600, 4620, 5280, 22176, 1, 4950, 68640, 110880, 133056, 576576, 1, 26586, 639870, 3141600, 3259872, 4036032, 17873856, 1, 234528, 10759056, 69263040, 105557760, 113008896, 142990848

Offset: 1

Peter Luschny, Mar 09 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -6,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134278.
Same partition product with length statistic is A049385.
Diagonal a(A000217) = A008548.
Row sum is A049412.

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

Cf. A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, 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}(-5*j - 1).

Offset corrected by Peter Luschny, Mar 14 2009

A049120 Row sums of triangle A049029.

Original entry on oeis.org

1, 6, 61, 871, 15996, 358891, 9509641, 290528316, 10051973371, 388433817091, 16579346005806, 774580047063901, 39313104018590221, 2153825039102763846, 126681355435102649161, 7961385691338995966371, 532402860878855993673036, 37746950872336992298209151

Offset: 1

Wolfdieter Lang

Generalized Bell numbers B(5,1;n).

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

Cf. Generalized Bell numbers B(m, 1, n): A049118 (m=3), A049119 (m=4), this sequence (m=5), A049412 (m=6).

With[{nn=20},CoefficientList[Series[Exp[-1+1/Surd[1-4x,4]]-1,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 10 2019 *)

E.g.f. exp(-1+1/(1-4*x)^(1/4))-1.
Representation of a(n) as the n-th moment of a positive function on positive half-axis (Stieltjes moment problem), in Maple notation: a(n)=int(x^n*exp(-1)*exp(-1/4*x)*(1/96*x*hypergeom([],[5/4, 3/2, 7/4, 2],1/1024*x)+ 1/8*4^(3/4)*x^(1/4)/Pi*2^(1/2)*GAMMA(3/4)*hypergeom([],[1/4, 1/2,3/4, 5/4],1/1024*x)+1/8*4^(1/2)*x^(1/2)/Pi^(1/2)*hypergeom([],[1/2, 3/4, 5/4,3/2],1/1024*x)+1/24*4^(1/4)*x^(3/4)/GAMMA(3/4)*hypergeom([],[3/4, 5/4, 3/2,7/4],1/1024*x))/x, x=0..infinity),n=1,2... . - Karol A. Penson, Dec 16 2007
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x)^5*d/dx. Cf. A000110, A000262, A049118 and A049119. - Peter Bala, Nov 25 2011
a(n) = (1/e) * (-4)^n * n! * Sum_{k>=0} binomial(-k/4,n)/k!. - Seiichi Manyama, Jan 17 2025

A049119 Row sums of triangle A035469.

Original entry on oeis.org

1, 5, 41, 465, 6721, 117941, 2433145, 57673281, 1543866945, 46052954821, 1514472783561, 54426342354385, 2121878761891201, 89187219264121525, 4020175011403931801, 193438800635132796161, 9895634072548245693825, 536284759396849853348101, 30691678336547328623916905

Offset: 1

Wolfdieter Lang

Generalized Bell numbers B(4,1;n).

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
P. Blasiak, K. A. Penson and A. I. Solomon, Combinatorial coherent states via normal ordering of bosons.

Cf. Generalized Bell numbers B(m, 1, n): A049118 (m=3), this sequence (m=4), A049120 (m=5), A049412 (m=6).

Drop[CoefficientList[Series[Exp[-1+1/(1-3*x)^(1/3)]-1,{x,0,19}],x]Range[0,19]!,1] (* Stefano Spezia, Mar 31 2025 *)

E.g.f.: exp(-1+1/(1-3*x)^(1/3))-1.
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x)^4*d/dx. Cf. A000110, A000262, A049118 and A049120. - Peter Bala, Nov 25 2011
a(n) = (1/e) * (-3)^n * n! * Sum_{k>=0} binomial(-k/3,n)/k!. - Seiichi Manyama, Jan 17 2025

A092084 Row sums of triangle A092082 (S2(7) Stirling2 generalization).

Original entry on oeis.org

1, 8, 113, 2283, 59956, 1937111, 74285023, 3296959548, 166209034083, 9380840313601, 585971815302336, 40131720225336433, 2990153392901281153, 240791249514701885728, 20839616039099721143561

Offset: 0

Wolfdieter Lang, Mar 19 2004

Generalized Bell numbers B(7,1;n).

Apparently the same as A072402 (apart from offset). - R. J. Mathar, Aug 27 2025

Cf. A092085 (alternating row sums), A049412 (generalized Bell numbers B(6, 1, n)).

a(n) = Sum_{m=1..n} A092082(n, m), for n>=1.

E.g.f.: exp(-1+(1-6*x)^(-1/6)) - 1.

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A049385 Triangle of numbers related to triangle A049375; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297...

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A134278 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6).

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A157396 A partition product of Stirling_2 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

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A049120 Row sums of triangle A049029.

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A049119 Row sums of triangle A035469.

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A092084 Row sums of triangle A092082 (S2(7) Stirling2 generalization).

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