A008279 -id:A008279 - cp's OEIS frontend

A090212 Alternating row sums of array A078741 ((3,3)-Stirling2).

1, -4, 73, -3241, 223546, -10884061, -5437091357, 4560715140638, -2741631069546683, 1315509914960956853, -135771066929217673256, -969783690708328561039261, 1943740128890758048004419957, -2140191682145533094039398047820

Offset: 1

Wolfdieter Lang, Dec 01 2003

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.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

Cf. A000587, A090211. A069223 (non-alternating sum, generalized Bell-numbers).

a[n_] := -Sum[(-1)^k FactorialPower[k, 3]^n/k!, {k, 2, Infinity}]*E; Array[a, 14] (* Jean-François Alcover, Sep 01 2016 *)

a(n) := sum( A078741(n, k)*(-1)^(k+1), k=3..3*n), n>=1. a(0) := -1 may be added.
a(n) = -sum(((-1)^k)*(fallfac(k, 3)^n)/k!, k=3..infinity)*exp(1), with fallfac(k, 3)=A008279(k, 3)=k*(k-1)*(k-2) and n>=1. This produces also a(0)=-1.
E.g.f. if a(0)=-1 is added: -exp(1)*(sum(((-1)^k)*exp(fallfac(k, 3)*x)/k!, k=3..infinity)+1/2). Similar to derivation on top of p. 4656 of the Schork reference.

A090213 Alternating row sums of array A090214 ((4,4)-Stirling2).

Original entry on oeis.org

1, -15, 1169, -154079, -148969375, 778633335441, -4003896394897551, 27901641934428560705, -268555885416357907647039, 3460225909437698652973995569, -56404253763542830420650221273263, 1050004356721541004548911018674177377

Offset: 1

Wolfdieter Lang, Dec 01 2003

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.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

Cf. A000587, A090211-2. A071379 (non-alternating sum, generalized Bell-numbers).

a[n_] := Sum[(-1)^k FactorialPower[k, 4]^n/k!, {k, 2, Infinity}]*E; Array[a, 12] (* Jean-François Alcover, Sep 01 2016 *)

a(n) := sum( A090214(n, k)*(-1)^k, k=4..4*n), n>=1. a(0) := 1 may be added.
a(n) = sum(((-1)^k)*(fallfac(k, 4)^n)/k!, k=4..infinity)*exp(1), with fallfac(k, 4)=A008279(k, 4)=k*(k-1)*(k-2)*(k-3) and n>=1. This produces also a(0)=1.
E.g.f. if a(0)=1 is added: exp(1)*(sum(((-1)^k)*exp(fallfac(k, 4)*x)/k!, k=4..infinity) + A000166(3)/3!). with the subfactorials A000166. A000166(3)/3!=1/3. Similar to derivation on top of p. 4656 of the Schork reference.

A090218 Alternating row sums of array A090216 (generalized Stirling2 array S_{5,5}(n,m)).

Original entry on oeis.org

1, -56, -29809, 326279119, -2175016082574, -74839638000014951, 12021284427301302745281, -1570241381612307786517290066, 198470943846200888426002717105781, 5344440525443920698933785031734661899, -41721146701452069718231186424275967809608724

Offset: 1

Wolfdieter Lang, Dec 01 2003

M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

a(n) = -sum(((-1)^k)*(fallfac(k, 5)^n)/k!, k=5..infinity)*exp(1), with fallfac(k, 5)=A008279(k, 5)=product(k+1-r, r=1..5) and n>=1. This produces also a(0)=-1.

E.g.f. if a(0)=-1 is added: -exp(1)*(sum(((-1)^k)*exp(fallfac(k, 5)*x)/k!, k=5..infinity) + 3/8). 3/8=A000166(4)/4! with the subfactorials A000166. Similar to the derivation on top of p. 4656 of the Schork reference.

A091553 Third column (k=6) sequence of array A090214 ((4,4)-Stirling2) divided by 72.

Original entry on oeis.org

1, 704, 300096, 113762304, 41644855296, 15075073327104, 5436979231850496, 1958506906364411904, 705205813266345885696, 253891292037560301256704, 91402929045514567230160896, 32905302125838589613523861504

Offset: 0

Wolfdieter Lang, Feb 13 2004

Cf. A089518 (third column of array (3, 3)-Stirling2 divided by 9).

a(n)= A090214(n+2, 6)/72, n>=0.
a(n)= (15*(6*5*4*3)^n - 10*(5*4*3*2)^n + (4*3*2*1)^n)/3!.
G.f.: (1+200*x)/product(1-fallfac(p, 4)*x, p=4..6), with fallfac(n, m) := A008279(n, m) (falling factorials).
a(n)= ((4!)^n)*(1-2*5^(n+1)+binomial(6, 2)^(n+1))/3!. From eq.12 of the Blasiak et al. reference given in A078740 with r=4=s, k=6.

A093908 Let f(k, n) be the product of n consecutive numbers beginning with k. Then a(n) is the least k > 1+n(n-1)/2 such that f(k, n) is a multiple of f(1+n(n-1)/2, n).

Original entry on oeis.org

2, 3, 8, 39, 52, 187, 204, 863, 773, 6621, 34038, 2404, 34440, 223097, 11976, 1106290, 1980047, 85119892, 15308072, 496820597, 2590416388, 1087065675, 4736428784, 1128909067, 242793786666, 2791304683100, 273924845940

Offset: 1

Amarnath Murthy, Apr 24 2004

nonn

f(k, n) = A008279(n+k-1, n). 1+n*(n-1)/2 = A000124(n-1). f(1+n*(n-1)/2, n) = A057003(n).

a(28) > 88*10^12.

			a(4) = 39 because 39*40*41*42 is divisible by 7*8*9*10. No
smaller set gives a product that is a multiple of 7*8*9*10.

Cf. A000124, A008279, A057003, A093909.

Edited and extended by David Wasserman, Apr 25 2007

A258213 Number of permutations of {1,2,3,...,n} such that no even numbers are adjacent.

Original entry on oeis.org

1, 1, 2, 6, 12, 72, 144, 1440, 2880, 43200, 86400, 1814400, 3628800, 101606400, 203212800, 7315660800, 14631321600, 658409472000, 1316818944000, 72425041920000, 144850083840000, 9560105533440000, 19120211066880000, 1491376463216640000, 2982752926433280000

Offset: 0

Ran Pan, May 23 2015

nonn

Cf. A008279, A010790.

a:= n-> (m-> m!^2*(m+1))(iquo(n+1, 2, 'r'))/(2-r):
seq(a(n), n=0..24);  # Alois P. Heinz, Feb 14 2024

T(n,k) = n!/(n-k)!; \\ A008279
a(n) = ceil(n/2)!*T(ceil(n/2)+1, n\2); \\ Michel Marcus, Nov 24 2022

a(n) = factorial(ceiling(n/2))*fallfac(ceiling(n/2)+1, floor(n/2)), with fallfac = A008279.
a(2n) = A010790(n), a(2n-1) = A010790(n)/2.
D-finite with recurrence: (4*(n-2)^2 + 24*n - 80)*a(n) + (16*n+24)*a(n-1) - (n+2)*n*((n-2)^2 + 8*n - 17)*a(n-2) = 0. - Georg Fischer, Nov 23 2022

A316773 Triangle read by rows: T(n,m) = Sum_{k=m+1..n} (n-1)!/(k-1)!binomial(2n-k-1, n-1)*E(k,m) where E(n,m) is Euler's triangle A173018, T(0,0) = 1, n >= m >= 0.

Original entry on oeis.org

1, 1, 0, 3, 1, 0, 19, 10, 1, 0, 193, 119, 23, 1, 0, 2721, 1806, 466, 46, 1, 0, 49171, 34017, 10262, 1502, 87, 1, 0, 1084483, 770274, 255795, 47020, 4425, 162, 1, 0, 28245729, 20429551, 7235853, 1539939, 193699, 12525, 303, 1, 0, 848456353, 621858526, 230629024, 54314242, 8273758, 755170, 34912, 574, 1, 0

Offset: 0

Yuriy Shablya, Sep 13 2018

T(n,m) is the number of labeled binary trees of size n with m ascents on the left branch.

			Triangle begins:
--------------------------------------------------------------------------
n\k|       0         1         2        3       4      5     6   7   8   9
------+-------------------------------------------------------------------
0 |         1
1 |         1         0
2 |         3         1         0
3 |        19        10         1        0
4 |       193       119        23        1       0
5 |      2721      1806       466       46       1      0
6 |     49171     34017     10262     1502      87      1     0
7 |   1084483    770274    255795    47020    4425    162     1   0
8 |  28245729  20429551   7235853  1539939  193699  12525   303   1  0
9 | 848456353 621858526 230629024 54314242 8273758 755170 34912 574  1  0

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Yuriy Shablya, Dmitry Kruchinin, Vladimir Kruchinin, Method for Developing Combinatorial Generation Algorithms Based on AND/OR Trees and Its Application, Mathematics (2020) Vol. 8, No. 6, 962.

Cf. A033184, A008279, A173018.

T := (n,m) -> `if`(n=0, 1, add((n-1)!/(k-1)!*binomial(2*n-k-1, n-1)*
combinat:-eulerian1(k, m), k = m+1..n)):
for n from 0 to 6 do seq(T(n, k), k=0..n) od; # Peter Luschny, Sep 04 2020

Table[Boole[n == 0] + Sum[(n - 1)!/(k - 1)!*Binomial[2 n - k - 1, n - 1]*Sum[(-1)^j*(m + 1 - j)^k*Binomial[k + 1, j], {j, 0, m}], {k, m + 1, n}], {n, 0, 8}, {m, 0, n}] // Flatten (* Michael De Vlieger, Sep 04 2020 *)

T(n,m):=if m>n then 0 else if n=0 then 1 else sum((n-1)!/(k-1)!*binomial(2*n-k-1,n-1)*sum((-1)^j*(m+1-j)^k*binomial(k+1,j),j,0,m),k,m+1,n);

E.g.f.: Sum_{n >= m >= 0} T(n, m)/n! * x^n * y^m = E(C(x),y) = (y-1)/(y-exp(C(x)*(y-1))), where E(x,y) is an e.g.f. for Euler's triangle A173018.

T(n,m) = Sum_{k = m+1..n} C(n,k)*E(k,m)*P(n,n-k), T(0,0)=1, where C(n,m) is the transposed Catalan's triangle A033184, E(n,m) is Euler's triangle A173018, and P(n,m) is the number of k-permutations of n A008279.

A344391 T(n, k) = binomial(n - k, k) * factorial(k), for n >= 0 and 0 <= k <= floor(n/2). Triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 6, 1, 5, 12, 6, 1, 6, 20, 24, 1, 7, 30, 60, 24, 1, 8, 42, 120, 120, 1, 9, 56, 210, 360, 120, 1, 10, 72, 336, 840, 720, 1, 11, 90, 504, 1680, 2520, 720, 1, 12, 110, 720, 3024, 6720, 5040, 1, 13, 132, 990, 5040, 15120, 20160, 5040

Offset: 0

Peter Luschny, May 17 2021

The antidiagonal representation of the falling factorials (A008279).

			[ 0] [1]
[ 1] [1]
[ 2] [1,  1]
[ 3] [1,  2]
[ 4] [1,  3,  2]
[ 5] [1,  4,  6]
[ 6] [1,  5, 12,   6]
[ 7] [1,  6, 20,  24]
[ 8] [1,  7, 30,  60,  24]
[ 9] [1,  8, 42, 120, 120]
[10] [1,  9, 56, 210, 360, 120]
[11] [1, 10, 72, 336, 840, 720]

Cf. A122852 (row sums).

Cf. A008279, A122851.

T := (n, k) -> pochhammer(n + 1 - 2*k, k):
seq(print(seq(T(n, k), k=0..n/2)), n = 0..11);

def T(n, k): return rising_factorial(n + 1 - 2*k, k)
def T(n, k): return (-1)^k*falling_factorial(2*k - n - 1, k)
def T(n, k): return binomial(n - k, k) * factorial(k)
print(flatten([[T(n, k) for k in (0..n//2)] for n in (0..11)]))

T(n, k) = RisingFactorial(n + 1 - 2*k, k).

T(n, k) = (-1)^k*FallingFactorial(2*k - n - 1, k).

A365638 Triangular array read by rows: T(n, k) is the number of ways that a k-element transitive tournament can occur as a subtournament of a larger tournament on n labeled vertices.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 8, 24, 24, 6, 64, 256, 384, 192, 24, 1024, 5120, 10240, 7680, 1920, 120, 32768, 196608, 491520, 491520, 184320, 23040, 720, 2097152, 14680064, 44040192, 55050240, 27525120, 5160960, 322560, 5040, 268435456, 2147483648, 7516192768, 11274289152, 7046430720, 1761607680, 165150720, 5160960, 40320

Offset: 0

Thomas Scheuerle, Sep 14 2023

A tournament is a directed digraph obtained by assigning a direction for each edge in an undirected complete graph. In a transitive tournament all nodes can be strictly ordered by their reachability.

			Triangle begins:
     1
     1,     1
     2,     4,     2
     8,    24,    24,     6
    64,   256,   384,   192,    24
  1024,  5120, 10240,  7680,  1920,  120

Paul Erdős, On a problem in graph theory, The Mathematical Gazette, 47: 220-223 (1963).

Cf. A002866, A006125, A052670, A095340, A103904, A008279, A117260, A379614 (row sums).

Cf. A122027, A224886, A259105, A350608, A350609, A350610.

T := (n, k) -> 2^(((n-1)*n - (k-1)*k)/2) * n! / (n-k)!:
seq(seq(T(n, k), k = 0..n), n = 0..8);  # Peter Luschny, Nov 02 2023

T(n, k) = binomial(n, k)*k!*2^(binomial(n, 2) - binomial(k, 2))

T(n, k) = binomial(n, k)*k!*2^(binomial(n, 2) - binomial(k, 2)).
T(n, 0) = A006125(n).
T(n, 1) = A095340(n).
T(n, 2) = A103904(n).
T(n, n) = n!.
T(n, n-1) = A002866(n-1).
T(n, n-2) = A052670(n).
T(n, k) = A008279(n, k) * A117260(n, k). - Peter Luschny, Dec 31 2024

A091739 Third column (k=7) sequence of array A090216 ((5,5)-Stirling2) divided by 600.

Original entry on oeis.org

1, 4440, 12715200, 33158592000, 84365452800000, 213181366579200000, 537634980016128000000, 1355141067314135040000000, 3415172150786516582400000000, 8606389816065144913920000000000

Offset: 0

Wolfdieter Lang, Feb 13 2004

Cf. A091553 (third column of array (4, 4)-Stirling2 divided by 72).

a(n)= A090216(n+2, 7)/600, n>=0.
a(n)= ((5!)^n)*(1-2*6^(n+1)+binomial(7, 2)^(n+1))/(2*5). From eq.12 of the Blasiak et al. reference given in A007840 with r=5=s, k=7.
a(n)= (21*(7*6*5*4*3)^n - 12*(6*5*4*3*2)^n + (5*4*3*2*1)^n)/10.
G.f.: (1+1080*x)/product(1-fallfac(p, 5)*x, p=5..7), with fallfac(n, m) := A008279(n, m) (falling factorials).

cp's OEIS Frontend

A090212 Alternating row sums of array A078741 ((3,3)-Stirling2).

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A090213 Alternating row sums of array A090214 ((4,4)-Stirling2).

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A090218 Alternating row sums of array A090216 (generalized Stirling2 array S_{5,5}(n,m)).

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A091553 Third column (k=6) sequence of array A090214 ((4,4)-Stirling2) divided by 72.

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A093908 Let f(k, n) be the product of n consecutive numbers beginning with k. Then a(n) is the least k > 1+n*(n-1)/2 such that f(k, n) is a multiple of f(1+n*(n-1)/2, n).

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A258213 Number of permutations of {1,2,3,...,n} such that no even numbers are adjacent.

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A316773 Triangle read by rows: T(n,m) = Sum_{k=m+1..n} (n-1)!/(k-1)!*binomial(2*n-k-1, n-1)*E(k,m) where E(n,m) is Euler's triangle A173018, T(0,0) = 1, n >= m >= 0.

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A344391 T(n, k) = binomial(n - k, k) * factorial(k), for n >= 0 and 0 <= k <= floor(n/2). Triangle read by rows.

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A365638 Triangular array read by rows: T(n, k) is the number of ways that a k-element transitive tournament can occur as a subtournament of a larger tournament on n labeled vertices.

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A093908 Let f(k, n) be the product of n consecutive numbers beginning with k. Then a(n) is the least k > 1+n(n-1)/2 such that f(k, n) is a multiple of f(1+n(n-1)/2, n).

A316773 Triangle read by rows: T(n,m) = Sum_{k=m+1..n} (n-1)!/(k-1)!binomial(2n-k-1, n-1)*E(k,m) where E(n,m) is Euler's triangle A173018, T(0,0) = 1, n >= m >= 0.