A008297 -id:A008297 - cp's OEIS frontend

A049411 Triangle read by rows, the Bell transform of n!*binomial(5,n) (without column 0).

1, 5, 1, 20, 15, 1, 60, 155, 30, 1, 120, 1300, 575, 50, 1, 120, 9220, 8775, 1525, 75, 1, 0, 55440, 114520, 36225, 3325, 105, 1, 0, 277200, 1315160, 730345, 112700, 6370, 140, 1, 0, 1108800, 13428800, 13000680, 3209745, 291060, 11130, 180, 1, 0, 3326400

Offset: 1

Wolfdieter Lang

Previous name was: A triangle of numbers related to triangle A049327.
a(n,1) = A008279(5,n-1). a(n,m) =: S1(-5; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A013988(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).
For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			Row polynomial E(3,x) = 20*x + 15*x^2 + x^3.
Triangle starts:
{  1}
{  5,    1}
{ 20,   15,   1}
{ 60,  155,  30,  1}
{120, 1300, 575, 50, 1}

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Peter Luschny, The Bell transform

Cf. A049327.

Row sums give A049428.

rows = 10;
a[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

# uses[bell_matrix from A264428]
# Adds 1,0,0,0,... as column 0 at the left side of the triangle.
bell_matrix(lambda n: factorial(n)*binomial(5, n), 8) # Peter Luschny, Jan 16 2016

a(n, m) = n!*A049327(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, nE.g.f. for m-th column: (((-1+(1+x)^6)/6)^m)/m!.

New name from Peter Luschny, Jan 16 2016

A049424 Triangle read by rows, the Bell transform of n!*binomial(4,n) (without column 0).

Original entry on oeis.org

1, 4, 1, 12, 12, 1, 24, 96, 24, 1, 24, 600, 360, 40, 1, 0, 3024, 4200, 960, 60, 1, 0, 12096, 40824, 17640, 2100, 84, 1, 0, 36288, 338688, 270144, 55440, 4032, 112, 1, 0, 72576, 2407104, 3580416, 1212624, 144144, 7056, 144, 1, 0, 72576, 14515200, 41791680

Offset: 1

Wolfdieter Lang

Previous name was: A triangle of numbers related to triangle A049326.
a(n,1) = A008279(4,n-1). a(n,m) =: S1(-4; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A011801(n,m). The monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			E.g., row polynomial E(3,x) = 12*x + 12*x^2 + x^3.
Triangle starts:
   1;
   4,   1;
  12,  12,   1;
  24,  96,  24,   1;
  24, 600, 360,  40,   1;

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Peter Luschny, The Bell transform

Cf. A049326.

Row sums give A049427.

rows = 10;
a[n_, m_] := BellY[n, m, Table[k! Binomial[4, k], {k, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

# uses[bell_matrix from A264428]
# Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
bell_matrix(lambda n: factorial(n)*binomial(2, n), 8) # Peter Luschny, Jan 16 2016

a(n, m) = n!*A049326(n, m)/(m!*5^(n-m));
a(n, m) = (5*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1;
a(n, m) = 0, nE.g.f. for m-th column: (((-1+(1+x)^5)/5)^m)/m!.

New name from Peter Luschny, Jan 16 2016

A086915 Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).

Original entry on oeis.org

2, 4, 4, 12, 24, 8, 48, 144, 96, 16, 240, 960, 960, 320, 32, 1440, 7200, 9600, 4800, 960, 64, 10080, 60480, 100800, 67200, 20160, 2688, 128, 80640, 564480, 1128960, 940800, 376320, 75264, 7168, 256, 725760, 5806080, 13547520, 13547520, 6773760, 1806336

Offset: 1

Vladeta Jovovic, Sep 24 2003

Also the Bell transform of A052849(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

The coefficients of n! * L_n(-2*x,-1), where n! * L_n(-x,-1) are the normalized, unsigned Laguerre polynomials of order -1 of A105278, also known as the Lah polynomials, which are also a shifted version of n! * L_n(-x,1). Cf. p. 8 of the Gross and Matytsin link. - Tom Copeland, Sep 30 2016

			Triangle begins:
   2;
   4,   4;
  12,  24,  8;
  48, 144, 96, 16;
  ...

G. C. Greubel, Rows n=1..100 of the triangle, flattened
D. Gross and A. Matytsin, Instanton induced large N phase transitions in two and four dimensional QCD, arXiv:hep-th/9404004, 1994.
Index entries for sequences related to Laguerre polynomials

Cf. A008297, A052897 (row sums), A059110, A079621, A105278.

[Factorial(n)*Binomial(n-1,k-1)*2^k/Factorial(k): k in [1..n], n in [1..10]]; // G. C. Greubel, May 23 2018

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ...) as column 0.
BellMatrix(n -> 2*(n+1)!, 9); # Peter Luschny, Jan 26 2016

Flatten[Table[n!/k! Binomial[n-1,k-1]2^k,{n,10},{k,n}]] (* Harvey P. Dale, May 25 2011 *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[2*(#+1)!&, rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

for(n=1,10, for(k=1, n, print1(n!/k!*binomial(n-1,k-1)*2^k, ", "))) \\ G. C. Greubel, May 23 2018

E.g.f.: exp(2*x*y/(1-x)).
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (-2)^k * A008297(n, k) = 2^k * A105278(n, k).
Sum_{k=1..n} T(n, k) = 2 * n! * Hypergeometric1F1([1-n], [2], -2) = 2*(n-1)! * LaguerreL(n-1, 1, -2) = A253286(n, 2). (End)

A111598 Lah numbers: a(n) = n!*binomial(n-1,7)/8!.

Original entry on oeis.org

1, 72, 3240, 118800, 3920400, 122316480, 3710266560, 111307996800, 3339239904000, 100919250432000, 3088129063219200, 96012739965542400, 3040403432242176000, 98228418580131840000, 3241537813144350720000

Offset: 8

Wolfdieter Lang, Aug 23 2005

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.

G. C. Greubel, Table of n, a(n) for n = 8..440

Column 8 of unsigned A008297 and A111596.
Column 7 of A111597.
Cf. A001113, A001620, A091725, A099285.

[Factorial(n-8)*Binomial(n,8)*Binomial(n-1,7): n in [8..35]]; // G. C. Greubel, May 10 2021

Table[(n-8)!*Binomial[n-1,7]*Binomial[n,8], {n,8,35}] (* G. C. Greubel, May 10 2021 *)

[factorial(n-8)*binomial(n,8)*binomial(n-1,7) for n in (8..35)] # G. C. Greubel, May 10 2021

E.g.f.: ((x/(1-x))^8)/8!.
a(n) = (n!/8!)*binomial(n-1, 8-1).
If we define f(n,i,x) = Sum_{k=i..n}(Sum_{j=i..k} (binomial(k,j)*Stirling1(n,k)* Stirling2(j,i)*x^(k-j) ) ) then a(n) = (-1)^n*f(n,8,-8), (n>=8). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=8} 1/a(n) = 61096*(gamma - Ei(1)) + 54544*e - 338732/5, where gamma = A001620, Ei(1) = A091725 and e = A001113.
Sum_{n>=8} (-1)^n/a(n) = 2107448*(gamma - Ei(-1)) - 1257760/e - 6080436/5, where Ei(-1) = -A099285. (End)

A247500 Triangle read by rows: T(n, k) = n!*binomial(n + 1, k)/(k + 1)!, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 12, 6, 1, 24, 60, 40, 10, 1, 120, 360, 300, 100, 15, 1, 720, 2520, 2520, 1050, 210, 21, 1, 5040, 20160, 23520, 11760, 2940, 392, 28, 1, 40320, 181440, 241920, 141120, 42336, 7056, 672, 36, 1, 362880, 1814400, 2721600, 1814400, 635040, 127008, 15120, 1080, 45, 1

Offset: 0

Peter Luschny, Oct 17 2014

An alternative definition would have been: (n-k)!*N(n,k) where N(n,k) are the little Narayana numbers A090181(n,k). This adds a first column (1,0,0,...) to the triangle and amounts to (Gamma(n)*Gamma(n+1))/(Gamma(k)*Gamma(k+1)*Gamma(n-k+2)). - Peter Luschny, Jun 18 2015
From Peter Bala, Sep 03 2023: (Start)
Let E(y) = Sum_{n >= 0} y^n/(n+1)!. Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence n!*(n+1)! as defined in Wang and Wang.
Let B(y) = Sum_{n >= 0} y^n/(n!*(n+1)!) = 1/sqrt(y)*BesselI(1,2*sqrt(y)). A generating function for the triangle is E(y)*B(x*y) = 1 + (1 + x)*y/(1!*2!) + (2 + 3*x + x^2)*y^2/(2!*3!) + (6 + 12*x + 6*x^2 + x^3)*y^3/(3!*4!) + .... Cf. A105278 with a generating function exp(y)*B(x*y).
The n-th power of this array has a generating function E(y)^n*B(x*y). In particular, the matrix inverse has a generating function B(x*y)/E(y). (End)

			Triangle begins:
                      1;
                   1,    1;
                2,    3,    1;
             6,   12,    6,    1;
         24,   60,   40,   10,    1;
     120,  360,  300,  100,   15,    1;
  720, 2520, 2520, 1050,  210,   21,    1;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
W. Wang and T. Wang, Generalized Riordan arrays, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Cf. A247499 (row sums), A008297.
Cf. A000142, A000217, A001710.
Cf. A204515 (central terms), A105278, A004736.
Cf. A090181, A001263.

a247500 n k = a247500_tabl !! n !! k
a247500_row n = a247500_tabl !! n
a247500_tabl = zipWith (zipWith div) a105278_tabl a004736_tabl
-- Reinhard Zumkeller, Oct 19 2014

/* triangle */ [[Factorial(n)/Factorial(k) * Binomial(n+2, k+1) /(n+2): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 18 2014

T := (n,k) -> ((k+1)*(n+1)*GAMMA(n+1)^2)/(GAMMA(k+2)^2*GAMMA(n-k+2));
A247500 := (n, k) -> (n!/(k+1)!)*binomial(n + 1, k):

Table[((k + 1) (n + 1) Gamma[n + 1]^2)/(Gamma[k + 2]^2*
Gamma[n - k + 2]), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jun 19 2015 *)

T(n, k) = ((k+1)*(n+1)*Gamma(n+1)^2)/(Gamma(k+2)^2 *Gamma(n-k+2)). (original name)
T(n, k) = (n!/k!)*C(n+2, k+1)/(n+2).
T(n, 0) = A000142(n).
T(n, n-1) = A000217(n).
T(n+1, 1) = A001710(n+2).
Sum_{k=0..n} T(n, k) = A247499(n).
L(n+1, k+1) = T(n-1, k)*P(n) for n>=1 and 0<=k<=n; here L(n,k) denote the unsigned Lah numbers and P(n) the pronic numbers. - Peter Luschny, Oct 18 2014
T(n,k) = A105278(n+1,k+1) / (n+1-k), k=0..n. - Reinhard Zumkeller, Oct 19 2014
From Peter Bala, May 24 2023: (Start)
Triangle equals A164652 * A008277 (assuming the same offset for the three triangles).
This is equivalent to the Stirling number identity Sum_{i = 0..n} (n+1)!/(i+1)!* binomial(n,i)*Stirling1(i+1,k) = (-1)^(n+k+1)*Stirling1(n+1,k) for n, k >= 0. (End)

Name updated by Peter Luschny, Jan 09 2022

A055924 Exponential transform of Stirling1 triangle A008275.

Original entry on oeis.org

1, -1, 2, 2, -6, 5, -6, 22, -30, 15, 24, -100, 175, -150, 52, -120, 548, -1125, 1275, -780, 203, 720, -3528, 8120, -11025, 9100, -4263, 877, -5040, 26136, -65660, 101535, -101920, 65366, -24556, 4140, 40320, -219168, 590620, -1009260, 1167348, -920808, 478842, -149040, 21147

Offset: 1

Wouter Meeussen, Christian G. Bower, Jul 06 2000

|a(n,k)| = number of sets of permutations of {1,...,n} with k total cycles.
From David Callan, Sep 20 2007: (Start)
|a(n,k)| = Stirling1(n, k) * Bell(k) counts the above sets of permutations. To see this, recall that Stirling1(n, k) is the number of permutations of [n]={1,...,n} with k cycles and Bell(k) is the number of set partitions of [k].
Given such a permutation and set partition, write the permutation in standard cycle form (smallest entry first in each cycle and first entries decreasing left to right). For example, with n=15 and k=6, {{10}, {6, 11}, {5, 7, 15}, {3, 13, 12, 8}, {2, 14, 9}, {1, 4}} is in this standard cycle form.
Then combine cycles as specified by the partition to form a set of lists. For example, the partition 156-24-3 would yield {{10, 2, 14, 9, 1, 4}, {6, 11, 3, 13, 12, 8}, {5, 7, 15}}. The original first entries are now the record left-to-right lows.
Finally, apply to each list the well known transformation that sends # record lows to # cycles. The example yields {{4, 14, 1, 2, 10, 9}, {13, 11, 3, 6, 8, 12}, {7, 15, 5}}. This is a bijection to sets of lists (i.e. permutations) with a total of k cycles, as required. (End)

			Triangle begins:
   1;
  -1,    2;
   2,   -6,   5;
  -6,   22, -30,   15;
  24, -100, 175, -150, 52;
  ...
|a(3,2)| = 6 because (12)(3), (12)|(3), (13)(2), (13)|(2), (23)(1), (23)|(1).

Row sums of |a(n,k)| give A000262.

Cf. A000110, A008275, A008297, A055925, A104150.

E.g.f.: exp((1+x)^y-1).

a(n,k) = Stirling1(n,k) * Bell(k). - Vladeta Jovovic, Feb 01 2003

A145118 Denominator polynomials for continued fraction generating function for n!.

Original entry on oeis.org

1, 1, 1, -1, 1, -2, 1, -4, 2, 1, -6, 6, 1, -9, 18, -6, 1, -12, 36, -24, 1, -16, 72, -96, 24, 1, -20, 120, -240, 120, 1, -25, 200, -600, 600, -120, 1, -30, 300, -1200, 1800, -720, 1, -36, 450, -2400, 5400, -4320, 720, 1, -42, 630, -4200, 12600, -15120

Offset: 0

Paul Barry, Oct 02 2008

Row sums are A056920. T(n,1) gives quarter squares A002620. T(n,2) appears to coincide with 2*A000241(n+1).

			Triangle begins:
1;
1;
1,  -1;
1,  -2;
1,  -4,   2;
1,  -6,   6;
1,  -9,  18,    -6;
1, -12,  36,   -24;
1, -16,  72,   -96,   24;
1, -20, 120,  -240,  120;
1, -25, 200,  -600,  600,  -120;
1, -30, 300, -1200, 1800,  -720;
1, -36, 450, -2400, 5400, -4320, 720;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A021010, A008297, A066667, A105278.

T:= (n, k)-> (-1)^k* binomial(iquo(n+1, 2),k) *binomial(iquo(n, 2), k)*k!:
seq (seq (T(n, k), k=0..iquo(n, 2)), n=0..16);  # Alois P. Heinz, Dec 04 2012

T(n,k) = (-1)^k C(floor((n+1)/2),k) * C(floor(n/2),k)*k!.

A176021 Triangle T(n, k) = 1 - (-1)^n*(n! + 1) + A176013(n, k) + A176013(n, n-k+1) read by rows.

Original entry on oeis.org

1, 1, 1, 1, -10, 1, 1, 72, 72, 1, 1, -528, -678, -528, 1, 1, 4770, 6780, 6780, 4770, 1, 1, -48025, -87568, -68458, -87568, -48025, 1, 1, 524384, 1287776, 947520, 947520, 1287776, 524384, 1, 1, -6169282, -19982590, -18010942, -10305790, -18010942, -19982590, -6169282, 1

Offset: 1

Roger L. Bagula, Apr 06 2010

Row sums are: 1, 2, -8, 146, -1732, 23102, -339642, 5519362, -98631416, 1926628022, ...

			Triangle begins as:
  1;
  1,        1;
  1,      -10,         1;
  1,       72,        72,         1;
  1,     -528,      -678,      -528,         1;
  1,     4770,      6780,      6780,      4770,         1;
  1,   -48025,    -87568,    -68458,    -87568,    -48025,         1;
  1,   524384,   1287776,    947520,    947520,   1287776,    524384,        1;
  1, -6169282, -19982590, -18010942, -10305790, -18010942, -19982590, -6169282, 1;

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A008297, A176013, A176022.

A176013:= func< n,k | (-1)^n*(Factorial(n)/(k*Factorial(k)))*Binomial(n-1, k-1)*Binomial(n, k-1) >;
[1 - (-1)^n*(Factorial(n) + 1) + A176013(n, k) + A176013(n, n-k+1) : k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 08 2021

A176013[n_, k_]:= (-1)^n*(n!/(k*k!))*Binomial[n-1, k-1]*Binomial[n, k-1];
T[n_, m_]:= 1 - (-1)^n*(n! + 1) + A176013[n, k] + A176013[n, n-k+1];
Table[T[n, k], {n, 12}, {k, n}]//Flatten

def A176013(n, k): return (-1)^n*(factorial(n)/(k*factorial(k)))*binomial(n-1, k-1)*binomial(n, k-1)
flatten([[1 - (-1)^n*(factorial(n) + 1) + A176013(n, k) + A176013(n, n-k+1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 08 2021

T(n, k) = 1 - (-1)^n*(n! + 1) + A176013(n, k) + A176013(n, n-k+1).

T(n, k) = 1 - (-1)^n*(n! + 1) + binomial(n+1, k)*( A008297(n, k) + A008297(n, n-k+1) )/(n+1). - G. C. Greubel, Feb 08 2021

Edited by G. C. Greubel, Feb 08 2021

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A180047 Coefficient triangle of the numerators of the (n-th convergents to) the continued fraction w/(1 + w/(2 + w/(3 + w/(...)))).

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A293125 Expansion of e.g.f.: exp(-x/(1+x)).

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A049411 Triangle read by rows, the Bell transform of n!*binomial(5,n) (without column 0).

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A049424 Triangle read by rows, the Bell transform of n!*binomial(4,n) (without column 0).

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A086915 Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).

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A111598 Lah numbers: a(n) = n!*binomial(n-1,7)/8!.

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A247500 Triangle read by rows: T(n, k) = n!*binomial(n + 1, k)/(k + 1)!, 0 <= k <= n.

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