A049374 -id:A049374 - cp's OEIS frontend

A049402 Row sums of triangle A049374.

1, 1, 7, 61, 649, 8245, 122215, 2069425, 39328465, 827226505, 19047582055, 475956135205, 12815133759385, 369605936607805, 11361372997850695, 370609338222772825, 12780705695068446625, 464412124831585889425, 17729002673226394402375, 709180766131239680070925

Offset: 0

Wolfdieter Lang

Alois P. Heinz, Table of n, a(n) for n = 0..415
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Column k=6 of A291709.

Cf. A049374, A049427.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+4)!/5!*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, j - 1]*(j + 4)!/5!*a[n - j], {j, 1, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

E.g.f. exp(p(x)) with p(x) := (1-(1-x)^5)/(5*(1-x)^5) (E.g.f. first column of A049374).
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A005011(k).
a(n) = (1/exp(1/5)) * (-1)^n * n! * Sum_{k>=0} binomial(-5*k,n)/(5^k * k!). (End)

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

A144356 Partition number array, called M31(6), related to A049374(n,m)= |S1(6;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 6, 1, 42, 18, 1, 336, 168, 108, 36, 1, 3024, 1680, 2520, 420, 540, 60, 1, 30240, 18144, 30240, 17640, 5040, 15120, 3240, 840, 1620, 90, 1, 332640, 211680, 381024, 493920, 63504, 211680, 123480, 158760, 11760, 52920, 22680, 1470, 3780, 126, 1, 3991680, 2661120

Offset: 1

Wolfdieter Lang Oct 09 2008, Oct 28 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31(6;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
Sixth member (K=6) in the family M31(K) of partition number arrays.
If M31(6;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(6)|:= A049374.

			[1];[6,1];[42,18,1];[336,168,108,36,1];[3024,1680,2520,420,540,60,1];...
a(4,3)= 108 = 3*|S1(6;2,1)|^2. The relevant partition of 4 is (2^2).

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

A049402 (row sums).

A144355 (M31(5) array).

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(6;j,1)|^e(n,k,j),j=1..n)= M3(n,k)*product(|S1(6;j,1)|^e(n,k,j),j=1..n) with |S1(6;n,1)|= A001725(n+4) = (n+4)!/5!, n>=1 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. M3(n,k)=A036040.

A134140 Alternating row sums of triangle A049374 (S1p(6)).

Original entry on oeis.org

1, 5, 25, 95, -275, -14755, -278795, -4134145, -49014215, -304537195, 7111142545, 397535340575, 12667519999525, 327297915798125, 7052174242084525, 109425656597938175, -88497453300450575, -107470322009554282075, -6297063330456696598775

Offset: 1

Wolfdieter Lang Oct 12 2007

Cf. A049402 (row sums of A049374).

a(n) = Sum_{m=1..n} A049374(n,m)*(-1)^(m-1), n>=1.

E.g.f.: 1-exp(-(1-(1-x)^5)/(5*(1-x)^5)). Cf. e.g.f. first column of A049374.

A001725 a(n) = n!/5!.

Original entry on oeis.org

1, 6, 42, 336, 3024, 30240, 332640, 3991680, 51891840, 726485760, 10897286400, 174356582400, 2964061900800, 53353114214400, 1013709170073600, 20274183401472000, 425757851430912000, 9366672731480064000, 215433472824041472000, 5170403347776995328000

Offset: 5

N. J. A. Sloane

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=6) ~ exp(-x)/x*(1 - 6/x + 42/x^2 - 336/x^3 + 3024/x^4 - 30240/x^5 + 332640/x^6 - 3991680/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

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).

Vincenzo Librandi, Table of n, a(n) for n = 5..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 265.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling. II, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 107-108 1963 1-77.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

a(n)= A049374(n-4), n >= 1 (first column of triangle). Cf. A049460, A051339. a(n)= A051338(n-5, 0)*(-1)^(n-1) (first unsigned column of triangle).

Cf. A245334, A000142.

a001725 = (flip div 120) . a000142 -- Reinhard Zumkeller, Aug 31 2014

[Factorial(n)/120: n in [5..25]]; // Vincenzo Librandi, Jul 20 2011

lst={};Do[AppendTo[lst, n!/5! ], {n, 5, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)
Range[5,30]!/120 (* Harvey P. Dale, Dec 20 2014 *)

a(n)=n!/120 \\ Charles R Greathouse IV, Jul 19 2011

E.g.f. if offset 0: 1/(1-x)^6.
a(n) = A173333(n,5). - Reinhard Zumkeller, Feb 19 2010
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(k+6)/(x*(k+6) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
G.f.: W(0)/(40*x^2) -1/(20*x^2) -1/(5*x) , where W(k) = 1 + 1/( 1 - x*(k+4)/( x*(k+4) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 21 2013
a(n) = A245334(n,n-5) / 6. - Reinhard Zumkeller, Aug 31 2014
E.g.f.: x^5 / (5! * (1 - x)). - Ilya Gutkovskiy, Jul 09 2021
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=5} 1/a(n) = 120*e - 325.
Sum_{n>=5} (-1)^(n+1)/a(n) = 45 - 120/e. (End)

More terms from Harvey P. Dale, Dec 20 2014

A157386 A partition product of Stirling_1 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 6, 1, 18, 42, 1, 144, 168, 336, 1, 600, 2940, 1680, 3024, 1, 4950, 33600, 35280, 18144, 30240, 1, 26586, 336630, 717360, 444528, 211680, 332640, 1, 234528, 4870992, 11313120, 10329984, 5927040, 2661120, 3991680

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -6,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144356.
Same partition product with length statistic is A049374.
Diagonal a(A000217(n)) = rising_factorial(6,n-1), A001725(n+4).
Row sum is A049402.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_1 Triangles.

Cf. A157385, A157384, A157383, A157400, A126074, A157391, A157392, A157393, A157394, A157395

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-2}(j-n-4).

A145357 Lower triangular array, called S1hat(6), related to partition number array A145356.

Original entry on oeis.org

1, 6, 1, 42, 6, 1, 336, 78, 6, 1, 3024, 588, 78, 6, 1, 30240, 6804, 804, 78, 6, 1, 332640, 62496, 8316, 804, 78, 6, 1, 3991680, 753984, 85176, 9612, 804, 78, 6, 1, 51891840, 8273664, 1021608, 94248, 9612, 804, 78, 6, 1, 726485760, 109118016, 11394432, 1157688, 102024

Offset: 1

Wolfdieter Lang, Oct 17 2008

If in the partition array M31hat(6):=A145356 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(6). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first columns are A001725(n+4), A145359, A145360,...

			Triangle begins:
  [1];
  [6,1];
  [42,6,1];
  [336,78,6,1];
  [3024,588,78,6,1];
  ...

Wolfdieter Lang, First 10 rows of the array and more.
Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Cf. A145358 (row sums).

a(n,m) = sum(product(|S1(6;j,1)|^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. |S1(6,n,1)|= A049374(n,1) = A001725(n+4) = (n+4)!/5!.

A134141 Generalized unsigned Stirling1 triangle, S1p(7).

Original entry on oeis.org

1, 7, 1, 56, 21, 1, 504, 371, 42, 1, 5040, 6440, 1295, 70, 1, 55440, 114520, 36225, 3325, 105, 1, 665280, 2116800, 983920, 135975, 7105, 147, 1, 8648640, 40884480, 26714800, 5199145, 398860, 13426, 196, 1, 121080960, 826338240, 735469280

Offset: 1

Wolfdieter Lang, Oct 12 2007

Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A092082(n, m) =: S2(7; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m, m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+6 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 05 2007
A triangle of numbers related to triangle A132166.
a(n,1)= A001730(n,5), n>=1. a(n,m)=: S1p(7; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n, m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n, m) (unsigned Lah numbers). S1p(3; n,m)= A046089(n,m), S1p(4; n,m)= A049352, S1p(5; n,m)= A049353(n,m), S1p(6; n,m)= A049374(n, m).
The Bell transform of factorial(n+6)/factorial(6). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			{1}; {7,1}; {56,21,1}; {504,371,42,1}; ... E.g. Row polynomial E(3,x)=56*x+21*x^2+x^3.
a(4,2)= 371 = 4*(7*8)+3*(7*7) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*7*8)=56 colored versions, e.g., ((1c1),(2c1,3c7,4c5)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 7 colors, c1..c7, can be chosen and the vertex labeled 4 with j=2 can come in 8 colors, e.g., c1..c8. Therefore there are 4*((1)*(1*7*8))=224 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*7)*(1*7))=147 such forests, e.g. ((1c1,3c4)(2c1,4c7)) or ((1c1,3c6)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 05 2007

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

First column A001730(n+5), n>=1.

Row sums A132164. Alternating row sums A132165.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+6)!/6!, 9); # Peter Luschny, Jan 27 2016

a[n_, m_] /; n >= m >= 1 := a[n, m] = (6*m + n - 1)*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}]][[1 ;; 39]] (* _Jean-François Alcover, Jun 01 2011, after formula *)
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[(# + 6)!/6! &, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: factorial(n+6)/factorial(6), 10) # Peter Luschny, Jan 18 2016

a(n, m) = n!*A132166(n, m)/(m!*6^(n-m)); a(n, m) = (6*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n

A145356 Partition number array, called M31hat(6).

Original entry on oeis.org

1, 6, 1, 42, 6, 1, 336, 42, 36, 6, 1, 3024, 336, 252, 42, 36, 6, 1, 30240, 3024, 2016, 1764, 336, 252, 216, 42, 36, 6, 1, 332640, 30240, 18144, 14112, 3024, 2016, 1764, 1512, 336, 252, 216, 42, 36, 6, 1, 3991680, 332640, 181440, 127008, 112896, 30240, 18144, 14112

Offset: 1

Wolfdieter Lang, Oct 17 2008, Oct 28 2008

Each partition of n, ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31hat(6;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
Sixth member (K=6) in the family M31hat(K) of partition number arrays.
If M31hat(6;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(6):= A145357.

			Triangle begins
  [1];
  [6,1];
  [42,6,1];
  [336,42,36,6,1];
  [3024,336,252,42,36,6,1];
  ...
a(4,3)= 36 = |S1(6;2,1)|^2. The relevant partition of 4 is (2^2).

Wolfdieter Lang, First 10 rows of the array and more.
Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Cf. A145358 (row sums).

Cf. A144890 (M31hat(5) array), A145357 (S1hat(6)).

a(n,k) = product(|S1(6;j,1)|^e(n,k,j),j=1..n) with |S1(6;n,1)| = A049374(n,1) = A001725(n+4) = [1,6,42,336,3024,30240,332640,...] = (n+4)!/5!, n>=1 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.

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A049402 Row sums of triangle A049374.

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A144356 Partition number array, called M31(6), related to A049374(n,m)= |S1(6;n,m)| (generalized Stirling triangle).

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A134140 Alternating row sums of triangle A049374 (S1p(6)).

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A001725 a(n) = n!/5!.

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A157386 A partition product of Stirling_1 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

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A145357 Lower triangular array, called S1hat(6), related to partition number array A145356.

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A134141 Generalized unsigned Stirling1 triangle, S1p(7).

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A145356 Partition number array, called M31hat(6).

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