A008279 -id:A008279 - cp's OEIS frontend

A090440 Generalized Stirling2 array (4,3).

1, 24, 36, 12, 1, 1440, 5760, 6120, 2520, 456, 36, 1, 172800, 1339200, 2808000, 2420640, 1025280, 232920, 29400, 2040, 72, 1, 36288000, 471744000, 1643846400, 2381702400, 1745755200, 721224000, 178941600, 27624960, 2689920, 163800, 6000

Offset: 1

Wolfdieter Lang, Dec 23 2003

The row lengths for this array are [1,4,7,10,13,16,...] = A016777(n-1), n>=1.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

W. Lang, First 6 rows.

Cf. A090438 (4, 2)-Stirling2.

Cf. A070531 (row sums), A091028 (alternating row sums).

ff[n_, k_] = Pochhammer[n - k + 1, k]; a[1, 3] = 1; a[n_, k_] := a[n, k] = Sum[Binomial[3, p]*ff[(n - 1 - p + k), 3 - p]*a[n - 1, k - p], {p, 0, 3} ]; a[n_ /; n < 2, ] = 0; Flatten[Table[a[n, k], {n, 1, 5}, {k, 3, 3 n}]] (* _Jean-François Alcover, Sep 01 2011, after given recursion *)

Recursion: a(n, k)=sum(binomial(3, p)*fallfac(n-1-p+k, 3-p)*a(n-1, k-p), p=0..3), n>=2, 3<=k<=3*n, a(1, 3)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=4, s=3. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+j-1, 3), j=1..n), p=3..k), n>=1, 3<=k<=3*n, else 0. From eq. (12) of the Blasiak et al. reference with r=4, s=3.

A091748 Generalized Bell numbers B_{6,2}.

Original entry on oeis.org

1, 43, 5083, 1160113, 432168721, 238012552651, 181520958432283, 182989529196234433, 235492729726705299073, 376560458072018837889931, 732162019709408940671604091, 1700645336651586566571229542193

Offset: 1

Wolfdieter Lang, Feb 27 2004

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A072019 ( B_{5, 2}).

a[n_] := Sum[Product[FactorialPower[k+4*(j-1), 2], {j, 1, n}]/k!, {k, 2, Infinity}]/E; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Sep 01 2016 *)

a(n)=sum(A091746(n, k), k=2..2*n)= sum((1/k!)*product(fallfac(k+4*(j-1), 2), j=1..n), k=2..infinity)/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=6, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.

A091749 Generalized Bell numbers B_{7,2}.

Original entry on oeis.org

1, 57, 9367, 3039037, 1631142633, 1306299636853, 1458563053824871, 2164056543968020185, 4116264432907357578961, 9762542731516508922640177, 28237035023990471230544779095, 97815632146487780258222172635029

Offset: 1

Wolfdieter Lang, Feb 27 2004

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A091748 (B_{6, 2}).

a[n_] := Sum[Product[FactorialPower[k+5*(j-1), 2], {j, 1, n}]/k!, {k, 2, Infinity}]/E; Array[a, 12] (* Jean-François Alcover, Sep 01 2016 *)

a(n)=sum(A091747(n, k), k=2..2*n)= sum((1/k!)*product(fallfac(k+5*(j-1), 2), j=1..n), k=2..infinity)/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=7, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.

A097662 a(n) = A002720(n) - 1.

Original entry on oeis.org

0, 1, 6, 33, 208, 1545, 13326, 130921, 1441728, 17572113, 234662230, 3405357681, 53334454416, 896324308633, 16083557845278, 306827170866105, 6199668952527616, 132240988644215841, 2968971263911288998, 69974827707903049153, 1727194482044146637520, 44552237162692939114281

Offset: 0

Ross La Haye, Sep 20 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A002720, A008279, A129833.

Main diagonal of A329655.

[Factorial(n)*Evaluate(LaguerrePolynomial(n), -1) -1: n in [0..40]]; // G. C. Greubel, Aug 11 2022

a := n -> hypergeom([-n, -n], [], 1) - 1:
seq(simplify(a(n)), n=0..26); # Peter Luschny, Oct 11 2016

Table[n!*LaguerreL[n,-1] -1, {n,0,40}] (* G. C. Greubel, Aug 11 2022 *)

[factorial(n)*laguerre(n, -1) -1 for n in (0..40)] # G. C. Greubel, Aug 11 2022

a(n) = Sum_{k=1..n} (n!^2 / k!*(n-k)!^2).
a(n) = Sum_{k=1..n} P(n, k)*C(n, k) where P(n,k), are the permutation coefficients A008279.
a(n) = n * A129833(n-1) for n>=1. - Peter Luschny, Oct 11 2016
From G. C. Greubel, Aug 11 2022: (Start)
E.g.f.: exp(x/(1-x))/(1-x) - exp(x).
Sum_{n >= 0} a(n)*x^n/(n!)^2 = (exp(x) -1)*BesselI(0, 2*sqrt(x)). (End)

A121757 Triangle read by rows: multiply Pascal's triangle by 1,2,6,24,120,720,... = A000142.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 18, 24, 1, 8, 36, 96, 120, 1, 10, 60, 240, 600, 720, 1, 12, 90, 480, 1800, 4320, 5040, 1, 14, 126, 840, 4200, 15120, 35280, 40320, 1, 16, 168, 1344, 8400, 40320, 141120, 322560, 362880, 1, 18, 216, 2016, 15120, 90720, 423360, 1451520

Offset: 0

Alford Arnold, Aug 19 2006

Row sums are 1,3,11,49,261,1631,... = A001339

a(n,k) = D(n+1,k+1) Array D in A253938 is part of a conjectured formula for F(n,p,r) that relates Dyck path peaks and returns. a(n,k) was discovered prior to array D. - Roger Ford, May 19 2016

			Row 6 is 1*1 5*2 10*6 10*24 5*120 1*720.
From _Vincenzo Librandi_, Dec 16 2012: (Start)
Triangle begins:
1,
1, 2,
1, 4,  6,
1, 6,  18,  24,
1, 8,  36,  96,   120,
1, 10, 60,  240,  600,  720,
1, 12, 90,  480,  1800, 4320,  5040,
1, 14, 126, 840,  4200, 15120, 35280,  40320,
1, 16, 168, 1344, 8400, 40320, 141120, 322560, 362880 etc.
(End)

Vincenzo Librandi, Rows n = 0..100, flattened
J. Goldman, J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 2-rook coefficients of k rooks on the 2xn board (all heights 2).
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007526 A000522, A005843 (2nd column), A028896 (3rd column).
Cf. A008279.
Cf. A008277, A132159 (mirrored).

a121757 n k = a121757_tabl !! n !! k
a121757_row n = a121757_tabl !! n
a121757_tabl = iterate
   (\xs -> zipWith (+) (xs ++ [0]) (zipWith (*) [1..] ([0] ++ xs))) [1]
-- Reinhard Zumkeller, Mar 06 2014

Flatten[Table[n!(k+1)/(n-k)!,{n,0,10},{k,0,n}]]  (* Harvey P. Dale, Apr 25 2011 *)

A000142(n)={ return(n!) ; } A007318(n,k)={ return(binomial(n,k)) ; } A121757(n,k)={ return(A007318(n,k)*A000142(k+1)) ; } { for(n=0,12, for(k=0,n, print1(A121757(n,k),",") ; ); ) ; } \\ R. J. Mathar, Sep 02 2006

a(n,k) = A007318(n,k)*A000142(k+1), k=0,1,..,n, n=0,1,2,3... - R. J. Mathar, Sep 02 2006

a(n,k) = A008279(n,k) * (k+1). a(n,k) = n!*(k+1)/(n-k)!. - Franklin T. Adams-Watters, Sep 20 2006

A145361 Characteristic partition array for partitions with parts 1 and 2 only.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Wolfdieter Lang, Oct 17 2008

Each partition of n, ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to 1 if the partition has parts 1 or 2 only and to 0 otherwise.
First member (K=1) in the family M31hat(-K) of partition number arrays.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
This array is array A144357 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144357/A036040'. E.g. a(4,3)= 1 = 3/3 = A144357(4,3)/A036040(4,3).
If M31hat(-1;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-1):= A145362 .

			Triangle begins:
  [1];
  [1,1];
  [0,1,1];
  [0,0,1,1,1];
  [0,0,0,0,1,1,1];
  ...
a(4,3)= 1 = S1(-1;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. A145363 (M31hat(-2)).

a(n,k) = product(S1(-1;j,1)^e(n,k,j),j=1..n) with S1(-1;n,1) = A008279(1,n-1) = [1,1,0,0,0,...], 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.

A145373 Lower triangular array, called S1hat(-5), related to partition number array A145372.

Original entry on oeis.org

1, 5, 1, 20, 5, 1, 60, 45, 5, 1, 120, 160, 45, 5, 1, 120, 820, 285, 45, 5, 1, 0, 1920, 1320, 285, 45, 5, 1, 0, 6600, 5420, 1945, 285, 45, 5, 1, 0, 9600, 23600, 7920, 1945, 285, 45, 5, 1, 0, 21600, 66600, 41100, 11045, 1945, 285, 45, 5, 1, 0, 14400, 189600, 151600, 53600, 11045

Offset: 1

Wolfdieter Lang, Oct 17 2008

If in the partition array M31hat(-5):=A145372 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-5). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first column is [1,5,20,60,120,120,0,0,0,...]= A008279(5,n-1), n>=1.

			Triangle begins:
  [1];
  [5,1];
  [20,5,1];
  [60,45,5,1];
  [120,160,45,5,1];
  ...

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. A145374 (row sums).

a(n,m) = sum(product(S1(-5;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(-5,n,1)= A008279(5,n-1) = [1,5,20,60,120,120,0,0,0,...], n>=1.

A349731 a(n) = -(-n)^n * FallingFactorial(1/n, n) for n >= 1 and a(0) = -1.

Original entry on oeis.org

-1, 1, 1, 10, 231, 9576, 623645, 58715280, 7547514975, 1270453824640, 271252029133449, 71635824470246400, 22929813173612997575, 8747686347650933760000, 3921812703436118765113125, 2041590849971133677650610176, 1221367737152989777782325269375, 832163138229382457228044554240000

Offset: 0

Peter Luschny, Dec 21 2021

sign

G. C. Greubel, Table of n, a(n) for n = 0..200

The main diagonal of A349971 for n >= 1.
The Stirling set counterpart is A318183.
Cf. A008279, A092985, A132393.

[-1,1] cat [Round(n^(n-1)*Gamma((n^2-1)/n)/Gamma((n-1)/n)): n in [2..30]]; // G. C. Greubel, Feb 22 2022

A349731 := n -> -add((-1)^(n-k)*Stirling1(n, n-k)*(-n)^k, k = 0..n):
seq(A349731(n), n = 0..17);

a[0] = -1; a[n_] := -(-n)^n * FactorialPower[1/n, n]; Array[a, 18, 0] (* Amiram Eldar, Dec 21 2021 *)

from sympy import ff
from fractions import Fraction
def A349731(n): return -1 if n == 0 else -(-n)**n*ff(Fraction(1,n),n) # Chai Wah Wu, Dec 21 2021

def a(n): return -(-n)^n*falling_factorial(1/n, n) if n > 0 else -1
print([a(n) for n in (1..17)])

a(n) = -(-1)^n*Sum_{k=0..n}[n, n-k]*(-n)^k, where [n, k] denotes the Stirling cycle numbers A132393(n, k).

A072678 Generalized Bell numbers B_{4,2}.

Original entry on oeis.org

1, 21, 1045, 93289, 12975561, 2581284541, 693347907421, 241253367679185, 105394372192969489, 56410454014314490981, 36271084122927079387941, 27567930377271475039277881, 24435533594428382909107147225

Offset: 1

Karol A. Penson, Jul 01 2002

nonn

Robert Israel, Table of n, a(n) for n = 1..220
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. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A090439 (alternating row sums of A090438).

f:= n -> simplify((2*n)!*hypergeom([2*n+1], [3], 1)/(2*exp(1))):
map(f, [$1..30]); # Robert Israel, May 23 2016

a[n_] := n*(2n-1)!*Hypergeometric1F1[2-2n, 3, -1]; Array[a, 30] (* Jean-François Alcover, Sep 01 2016 *)

a(n) = (2*n)!*hypergeom([2*n+1], [3], 1)/(2*exp(1)), n=1, 2, ... Special values of the confluent hypergeometric function 1F1.
a(n) = sum(A090438(n, k), k=2..2*n)= sum((1/k!)*product(fallfac(k+(j-1)*(4-2), 2), j=1..n), k=1..infinity)/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=4, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.
8*n*(2*n-1)*(2*n+1)*(n+1)^2*(n+3)*(n+2)*a(n)+(2*(n+1))*(8*n^3+32*n^2+42*n+13)*a(n+1)*(n+3)*(n+2)-(8*n^2+38*n+51)*(n+3)*(n+2)*a(n+2)+(n+3)*(n+2)*a(n+3) = 0. - Robert Israel, May 23 2016
a(n) = A052852(2*n-1). - Mark van Hoeij, Sep 05 2022

Edited by Wolfdieter Lang, Dec 23 2003

A076014 Triangle in which m-th entry of n-th row is m^(n-1).

Original entry on oeis.org

1, 1, 2, 1, 4, 9, 1, 8, 27, 64, 1, 16, 81, 256, 625, 1, 32, 243, 1024, 3125, 7776, 1, 64, 729, 4096, 15625, 46656, 117649, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721

Offset: 1

Wolfdieter Lang, Oct 02 2002

This becomes triangle A009998(n-1, m-1), n >= m >= 1, if the m-th column entries are divided by m^(m-1).
Row sums give A076015. The m-th column (without leading zeros) gives (m^(m-1)) powers of m, m >= 1.
T(n,m) is the number of functions f:[n-1]->[(n-1)m] such that f(x)=k*x for some positive integer k <= m. Since there exactly m choices for each of the (n-1) images under f, we obtain T(n,m) = m^(n-1). - Dennis P. Walsh, Feb 27 2013
T(n+1,m+1) = (m+1)^n is the number of partial functions from an n-element set to an m-element set, n >= m >= 0. - Mohammad K. Azarian, Jun 28 2021

			For example, T(3,2)=4 since there are exactly 4 functions f from {1,2} to {1,2,3,4} that satisfy f(x)=x or f(x)=2x. If we specify each function by the ordered pair (f(1),f(2)), the four functions are (1,2), (1,4), (2,2), and (2,4). - _Dennis P. Walsh_, Feb 27 2013
Triangle begins:
  1;
  1,   2;
  1,   4,    9;
  1,   8,   27,    64;
  1,  16,   81,   256,   625;
  1,  32,  243,  1024,  3125,   7776;
  1,  64,  729,  4096, 15625,  46656, 117649;
  1, 128, 2187, 16384, 78125, 279936, 823543, 2097152;
  ...

Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Theorem 2.5, p. 132.

Cf. A009998, A008279, A008277 (Stirling2).

Cf. A089072.

seq(seq(m^(n-1),m=1..n),n=1..20); # Dennis P. Walsh, Feb 27 2013

Table[m^(n-1),{n,10},{m,n}]//Flatten (* Harvey P. Dale, May 27 2017 *)

T(n, m) = m^(n-1), n >= m >= 1, otherwise 0.
G.f. for m-th column: (m^(m-1))(x^m)/(1-m*x), m >= 1.
a(n,m) = Sum_{p=1..m} Stirling2(n,p)*A008279(m-1, p-1), n >= m >= 1, otherwise 0.

cp's OEIS Frontend

A090440 Generalized Stirling2 array (4,3).

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A091748 Generalized Bell numbers B_{6,2}.

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A091749 Generalized Bell numbers B_{7,2}.

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A097662 a(n) = A002720(n) - 1.

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A121757 Triangle read by rows: multiply Pascal's triangle by 1,2,6,24,120,720,... = A000142.

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A145361 Characteristic partition array for partitions with parts 1 and 2 only.

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A145373 Lower triangular array, called S1hat(-5), related to partition number array A145372.

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A349731 a(n) = -(-n)^n * FallingFactorial(1/n, n) for n >= 1 and a(0) = -1.

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