A008279 -id:A008279 - cp's OEIS frontend

A076482 Triangle with T(n,k)=n!*(k-1)^k/k! where 1<=k<=n.

0, 0, 1, 0, 3, 8, 0, 12, 32, 81, 0, 60, 160, 405, 1024, 0, 360, 960, 2430, 6144, 15625, 0, 2520, 6720, 17010, 43008, 109375, 279936, 0, 20160, 53760, 136080, 344064, 875000, 2239488, 5764801, 0, 181440, 483840, 1224720, 3096576, 7875000, 20155392, 51883209, 134217728

Offset: 1

Henry Bottomley, Oct 14 2002

			Rows start
  0;
  0,   1;
  0,   3,   8;
  0,  12,  32,   81;
  0,  60, 160,  405, 1024;
  0, 360, 960, 2430, 6144, 15625;
  ...

Seiichi Manyama, Rows n = 1..140, flattened

Row sums are A076483.

Cf. A001710, A007778, A008279.

Table[n! (k-1)^k/k!,{n,0,10},{k,n}]//Flatten (* Harvey P. Dale, Nov 28 2019 *)

T(n,k) = n*T(n, k-1) = A007778(k-1)*A008279(n,n-k) starting with T(n,n) = (n-1)^n = A007778(n-1).

A078738 Generalized Bell numbers B_{3,2}(n).

Original entry on oeis.org

1, 13, 355, 16333, 1121881, 106708921, 13354028563, 2118817455385, 414426460442833, 97746679844312581, 27311169061720393411, 8908525371578726747173, 3350963996380181114090665, 1438463413778071631322236593, 698374517715612292764726380851

Offset: 1

N. J. A. Sloane, Dec 21 2002

Vaclav Kotesovec, Table of n, a(n) for n = 1..240
P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/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.

B_{1, 1} = A000110, B_{2, 1} = A000262, B_{3, 1} = A020556 and B_{3, 3} = A069223. Row sums of A078740.

Alternating row sums A090437.

a[n_] := (n+1)*n!^2*Sum[(-1)^k*HypergeometricPFQ[{2-k, n+1, n+2}, {2, 3}, 1]/(2*(k-2)!), {k, 2, 2n}]; Array[a, 13] (* Jean-François Alcover, Sep 01 2015 *)
Table[Sum[(n + k)!*(n + k + 1)!/(k!*(k + 1)!*(k + 2)!), {k, 0, Infinity}]/E, {n, 1, 20}] (* Vaclav Kotesovec, Jul 27 2018 *)

nmax = 20; p = floor(3*nmax*log(nmax)); default(realprecision, p);
for(n=1, nmax, print1(round(exp(-1)*suminf(k=0, (n+k)!*(n+k+1)!/(k!*(k+1)!*(k+2)!))), ", ")) \\ G. C. Greubel and Vaclav Kotesovec, Jul 28 2018

a(n) = Sum_{k=2..2*n} A078740(n, k) = Sum_{k=1..infinity} (1/k!)*Product_{j=1..n}(fallfac(k+(j-1)*(3-2), 2))/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=3, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.
a(n) = Sum_{k>=0} ((n+k)!*(n+k+1)!/(k!*(k+1)!*(k+2)!))/exp(1), n>=1. From eq.(40) of the Blasiak et al. reference. [corrected by Vaclav Kotesovec, Jul 27 2018]
E.g.f. for a(n)/n! with a(0)=(exp(1)-1)/exp(1) added: Sum_{k>=0} (hypergeom([k+2, k+1], [1], z)/(k+2)!)/exp(1). From eq. (41) of the Blasiak et al. reference.

Edited by Wolfdieter Lang, Dec 23 2003

A090221 Array used for numerators of g.f.s for column sequences of array A090214 ((4,4)-Stirling2).

Original entry on oeis.org

1, 96, 72, 14400, 16, 38400, 3456000, 1, 27000, 22104000, 1270080000, 7200, 34905600, 16111872000, 682795008000, 856, 21154176, 48248363520, 15279164006400, 516193026048000, 48, 6064128, 54644474880, 78083415244800

Offset: 4

Wolfdieter Lang, Dec 01 2003

The row length sequence for this array is A037915(k-4)= floor(3*(k-4)/4)+1, k>=4: [1, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, ...].

The g.f. G(k,x) for the k-th column (with leading zeros) of array A090214 is given there. The recurrence is G(k,x) = x*sum(binomial(k-r,4-r)*fallfac(4,4-r)*G(k-r,x),r=1..4))/(1-fallfac(k,4)*x), k>=4, with inputs G(k,x)=0 for k=1,2,3 and G(4,x)=x/(1-4!*x); where fallfac(n,m) := A008279(n,m) (falling factorials with fallfac(n,0) := 1). Computed from the Blasiak et al. reference, eqs. (20) and (21) with r=4: recurrence for S_{4,4}(n,k).

			[1]; [96]; [72,14400]; [16,38400,3456000]; [1,27000,22104000,1270080000]; ...
G(5,x)/x^2 = 96/((1-4!*x)*(1-5*4*3*2*x)). kmax(5)=0, hence P(5,x)=a(5,0)=96; x^2 from x^ceiling(5/4).

W. Lang, First 8 rows.

a(k, n) from: sum(a(k, n)*x^n, n=0..kmax(k)) = G(k, x)* product(1-fallfac(p, 4)*x, p=4..k)/x^ceiling(k/4), k>=4, with G(k, x) defined from the recurrence given above and kmax(k) := A057353(k-4)= floor(3*(k-4)/4)= A037915(k-4)-1.

A091757 Generalized Bell numbers B_{8,2}.

Original entry on oeis.org

1, 73, 15945, 6993073, 5124715761, 5641397595321, 8700819552421753, 17898786381229403105, 47345052327747786859873, 156535091017683923932912041, 632460052562874236182866885161

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. A091749 (B_{7, 2}).

a[n_] := Sum[Product[FactorialPower[k+6*(j-1), 2], {j, 1, n}]/k!, {k, 2, Infinity}]/E; Array[a, 11] (* Jean-François Alcover, Sep 01 2016 *)

a(n)=sum(A092077(n, k), k=2..2*n)= sum((1/k!)*product(fallfac(k+6*(j-1), 2), j=1..n), k=2..infinity)/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=8, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.

A248716 Numerator of (1/e)Sum_{k>=0} (1/k!)(Sum_{j=0..k} j^n).

Original entry on oeis.org

2, 3, 17, 27, 293, 791, 10583, 25685, 448303, 251411, 4503535, 6331107, 4436875097, 3335427631, 19619696071, 75379875277, 3019260651391, 16773385986619, 3047463007411973, 2732480436961811, 398377271835431771, 173581842021095897, 1716900426430701553, 35001773773285490879, 6684326532123939298051

Offset: 0

Richard R. Forberg, Dec 28 2014

Denominators are given by A130190, which in that entry are associated with the denominators of the z-sequence for a certain Sheffer matrix (triangle), but also apply here. See also the formula given there for the denominator of a certain sum involving the Stirling2 numbers.
Note that a(0) = 2 uses 0^0 := 1. For n >= 1 use triangle A079618 and the formula (1/e)*Sum_{k>=0} ((k+1)^n)/k! = Bell(n+1) = A000110(n+1). - Wolfdieter Lang, Feb 03 2015
If instead of Sum_{j=0..k} j^n one uses the sum with falling factorials, namely F(k, n) := Sum_{j=0..k} A008279(j, n) = A008279(k+1, n+1)/(n+1) the result for the rationals R(n) = (1/e)*Sum_{k>=0} (1/k!)*F(k, n) becomes very simple, namely R(n) = (n+2)/(n+1), n >= 0. - Wolfdieter Lang, Feb 03 2015

			Terms up to n = 10, with denominators, are 2/1, 3/2, 17/6, 27/4, 293/15, 791/12, 10583/42, 25685/24, 448303/90, 251411/10, 4503535/33, ... .
From _Wolfdieter Lang_, Feb 03 2015: (Start)
With triangle A079618, A064538 and the Bell numbers A000110 the rationals r(n) are:
n=4: (1/30)*(-1*1 + 0*2 + 10*5 + 15*15 + 6*52) = 293/15.
n=9: (1/20)*(0*1 + (-3)*2 +  0*5 + 10*15 + 0*52 + (-14)*203 + 0*877 + 15*4140 + 10*21147 + 2*115975) = 251411/10.
(End)

G. C. Greubel, Table of n, a(n) for n = 0..251

Cf. A000110, A064538, A079618, A130190.

Numerator[Table[(1/Exp[1])*Sum[Sum[j^n/k!, {j, 0, k}], {k, 0, Infinity}],
{n, 0, 100}]]

a(n) = numerator(r(n)) with the rationals r(n) = (1/e)*Sum_{k>=0} (1/k!)*(Sum_{j=0..k} j^n), n >= 0, where 0^0 := 1.

a(n) = numerator(r(n)), with r(n) = (1/A064538(n))*Sum_{k=0..n} T(n+1,k+1)*Bell(k+1), with T(n,k) = A079618(n,k) and Bell(n) = A000110(n), for n >= 1. a(0) = 2 using 0^0 := 1. See comments above. - Wolfdieter Lang, Feb 03 2015

Edited. Comment and formula rewritten. Cross references added. - Wolfdieter Lang, Feb 03 2015

A288853 Triangle read by rows: T(n,k) is the number of surjective linear mappings from an n-dimensional vector space over F_2 onto a k-dimensional vector space, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 1, 7, 42, 168, 1, 15, 210, 2520, 20160, 1, 31, 930, 26040, 624960, 9999360, 1, 63, 3906, 234360, 13124160, 629959680, 20158709760, 1, 127, 16002, 1984248, 238109760, 26668293120, 2560156139520, 163849992929280, 1, 255, 64770, 16322040, 4047865920, 971487820800, 217613271859200, 41781748196966400, 5348063769211699200

Offset: 0

Geoffrey Critzer, Jun 18 2017

The (q = 2) analog of A008279.
A022166(m,k)*T(n,k) is the number of m X n matrices over F_2 that have rank k.
a(n) is the number of n X n matrices over F_2 in Green's R class containing A where rank(A) = k. - Geoffrey Critzer, Oct 05 2022

			  1;
  1,  1;
  1,  3,   6;
  1,  7,  42,   168;
  1, 15, 210,  2520,  20160;
  1, 31, 930, 26040, 624960, 9999360;
  ...

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Jeremy L. Martin, Lecture Notes on Algebraic Combinatorics, 2010-2023, Example 2.3.6.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Wikipedia, Green's relations.

Columns k=0-10 give: A000012, A000225, 6*A006095, 168*A006096, 20160*A006097, 9999360*A006110, 20158709760*A022189, 163849992929280*A022190, 5348063769211699200*A022191, 699612310033197642547200*A022192, 366440137299948128422802227200*A022193.
Main diagonal gives A002884.
Cf. A022166.

Table[Table[Product[q^n - q^i, {i, 0, k - 1}] /. q -> 2, {k, 0, n}], {n, 0,8}] // Grid

T(n,k) = Product_{j=0..k-1} (2^n - 2^j).
T(n,k) = A002884(k)*A022166(n,k).
Let g_m(x) = Sum_{n>=0} (2^m*x)^n/A005329(n) and e(x) = Sum_{n>=0} x^n/A005329(n).  Then Sum_{k>=0} T(n,k)*x^k/A005329(k) = g_n(x)/e(x). - Geoffrey Critzer, Jun 01 2024

A338993 Triangle read by rows: T(n,k) is the number of k-permutations of {1,...,n} that form a nontrivial arithmetic progression, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 6, 2, 4, 12, 4, 2, 5, 20, 8, 4, 2, 6, 30, 12, 6, 4, 2, 7, 42, 18, 10, 6, 4, 2, 8, 56, 24, 14, 8, 6, 4, 2, 9, 72, 32, 18, 12, 8, 6, 4, 2, 10, 90, 40, 24, 16, 10, 8, 6, 4, 2, 11, 110, 50, 30, 20, 14, 10, 8, 6, 4, 2, 12, 132, 60, 36, 24, 18, 12, 10, 8, 6, 4, 2

Offset: 1

Marcel K. Goh, Nov 17 2020

The step size ranges from 1 to floor((n-1)/(k-1)) and for each r, there are 2*(n-(k-1)*r) possible ways to form a progression.

Proof can be found in Lemma 1 of Goh and Zhao (2020).

			Triangle T(n,k) begins:
  n/k  1   2   3   4   5   6   7   8   9  10  11  12 ...
   1   1
   2   2   2
   3   3   6   2
   4   4  12   4   2
   5   5  20   8   4   2
   6   6  30  12   6   4   2
   7   7  42  18  10   6   4   2
   8   8  56  24  14   8   6   4   2
   9   9  72  32  18  12   8   6   4   2
  10  10  90  40  24  16  10   8   6   4   2
  11  11 111  50  30  20  14  10   8   6   4   2
  12  12 132  60  36  24  18  12  10   8   6   4   2
  ...
For n=4 and k=3 the T(4,3)=4 permutations are 123, 234, 321, and 432.

M. K. Goh and R. Y. Zhao, Arithmetic subsequences in a random ordering of an additive set, arXiv:2012.12339 [math.CO], 2020.

Cf. A008279.

T[n_,k_]:=If[k==1, n,Sum[2(n-(k-1)r),{r,Floor[(n-1)/(k-1)]}]]; Flatten[Table[T[n,k],{n,12},{k,n}]] (* Stefano Spezia, Nov 17 2020 *)

T(n,k) = if (k==1, n, sum(r=1, (n-1)\(k-1), 2*(n-(k-1)*r))); \\ Michel Marcus, Sep 08 2021

T(n,k) = n, if k=1; Sum_{r=1..floor((n-1)/(k-1))} 2*(n-(k-1)*r), if 2 <= k <= n.

T(n,k) = 2*n*f - (k-1)*(f^2 + f), where f = floor((n-1)/(k-1)), for 2 <= k <= n.

A349280 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k rounds; n >= 2, 1 <= k <= floor(n/2).

Original entry on oeis.org

2, 3, 8, 12, 30, 60, 144, 330, 120, 840, 2100, 1260, 5760, 15344, 11760, 1680, 45360, 127008, 113400, 30240, 403200, 1176120, 1169280, 428400, 30240, 3991680, 12054240, 13000680, 5821200, 831600, 43545600, 135508032, 155923680, 80415720, 16632000, 665280

Offset: 2

Steven Finch, Nov 13 2021

A round means the same as a directed ring or circle.

			Triangle starts:
[2]     2;
[3]     3;
[4]     8,     12;
[5]    30,     60;
[6]   144,    330,    120;
[7]   840,   2100,   1260;
[8]  5760,  15344,  11760,  1680;
[9] 45360, 127008, 113400, 30240;
...
For n = 4, there are 8 ways to make one round and 12 ways to make two rounds.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)

Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Row sums give A066166 (Stanley's children's game).
Column 1 gives A001048.
Right border element of row n is A001813(n/2) = |A067994(n)| for even n.
Cf. A008279, A265609, A331327.

ser := series((1 - x)^(-x*t), x, 20): xcoeff := n -> coeff(ser, x, n):
T := (n, k) -> n!*coeff(xcoeff(n), t, k):
seq(seq(T(n, k), k = 1..iquo(n,2)), n = 2..12); # Peter Luschny, Nov 13 2021
# second Maple program:
A349280 := (n,k) -> binomial(n,k)*k!*abs(Stirling1(n-k,k)):
seq(print(seq(A349280(n,k), k=1..iquo(n,2))), n=2..12); # Mélika Tebni, May 03 2023

f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t), {x, 0, n}, {t, 0, k}]
Table[f[k, n], {n, 2, 12}, {k, 1, Floor[n/2]}]

G.f.: (1 - x)^(-x*t).
T(n, k) = binomial(n, k)*k!*|Stirling1(n-k, k)|. - Mélika Tebni, May 03 2023
The above formula can also be written as T(n, k) = A008279(n, k)*A331327(n, k) or as T(n, k) = A265609(n + 1, k)*A331327(n, k). - Peter Luschny, May 03 2023

A387126 Triangle read by rows: T(n, k) = (n! / (n - k)!) * Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.

Original entry on oeis.org

1, 1, 1, 2, 4, 4, 6, 18, 36, 36, 12, 48, 144, 288, 288, 60, 300, 1200, 3600, 7200, 7200, 360, 2160, 10800, 43200, 129600, 259200, 259200, 2520, 17640, 105840, 529200, 2116800, 6350400, 12700800, 12700800, 5040, 40320, 282240, 1693440, 8467200, 33868800, 101606400, 203212800, 203212800

Offset: 0

Peter Luschny, Aug 18 2025

			Triangle begins:
  [0]    1;
  [1]    1,     1;
  [2]    2,     4,      4;
  [3]    6,    18,     36,     36;
  [4]   12,    48,    144,    288,     288;
  [5]   60,   300,   1200,   3600,    7200,    7200;
  [6]  360,  2160,  10800,  43200,  129600,  259200,   259200;
  [7] 2520, 17640, 105840, 529200, 2116800, 6350400, 12700800, 12700800;

Cf. A007947 (radical), A008279, A048803 (column 0), A277174 (main diagonal).

A387126 := (n, k) -> mul(NumberTheory:-Radical(j), j = 1..n) * n! / (n - k)!:

A387126[n_, k_] := Pochhammer[n-k+1, k] Times @@ ResourceFunction["IntegerRadical"][Range[1, n]];
Table[A387126[n, k], {n, 0, 8}, {k, 0, n}]  // Flatten

T(n, k) = A048803(n) * A008279(n, k).

A071417 Triangle of expected coupon collection numbers rounded up; i.e., if aiming to collect part of a set of n coupons, the expected number of random coupons required to receive first the set with exactly k missing.

Original entry on oeis.org

0, 1, 0, 3, 1, 0, 6, 3, 1, 0, 9, 5, 3, 1, 0, 12, 7, 4, 3, 1, 0, 15, 9, 6, 4, 3, 1, 0, 19, 12, 8, 6, 4, 3, 1, 0, 22, 14, 10, 8, 6, 4, 3, 1, 0, 26, 17, 12, 9, 7, 5, 4, 3, 1, 0, 30, 20, 15, 11, 9, 7, 5, 4, 3, 1, 0, 34, 23, 17, 14, 11, 9, 7, 5, 4, 3, 1, 0, 38, 26, 20, 16, 13, 10, 8, 7, 5, 4, 3, 1, 0, 42

Offset: 0

Henry Bottomley, May 29 2002

			Rows start
0;
1,0;
3,1,0;
6,3,1,0;
9,5,3,1,0;
etc.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150).

Cf. A060293 (left hand column), A067176.

Table[Ceiling[n Sum[1/j, {j, k + 1, n}]], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 30 2017 *)

a(n, k) = ceiling(n*Sum_{j=k+1..n} 1/j) = ceiling(A067176(n, k)*k!/(n-1)!) = ceiling(A008279(n, n-k)*Sum_{j>=n-k} j*A008277(j-1, n-k-1)/n^j).

cp's OEIS Frontend

A076482 Triangle with T(n,k)=n!*(k-1)^k/k! where 1<=k<=n.

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A078738 Generalized Bell numbers B_{3,2}(n).

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A090221 Array used for numerators of g.f.s for column sequences of array A090214 ((4,4)-Stirling2).

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A091757 Generalized Bell numbers B_{8,2}.

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A248716 Numerator of (1/e)*Sum_{k>=0} (1/k!)*(Sum_{j=0..k} j^n).

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A288853 Triangle read by rows: T(n,k) is the number of surjective linear mappings from an n-dimensional vector space over F_2 onto a k-dimensional vector space, n>=0, 0<=k<=n.

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A338993 Triangle read by rows: T(n,k) is the number of k-permutations of {1,...,n} that form a nontrivial arithmetic progression, 1 <= k <= n.

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A349280 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k rounds; n >= 2, 1 <= k <= floor(n/2).

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A248716 Numerator of (1/e)Sum_{k>=0} (1/k!)(Sum_{j=0..k} j^n).