A034841 -id:A034841 - cp's OEIS frontend

A096127 a(n) is the largest k such that (n^2)!/(n!)^k is an integer.

3, 4, 5, 6, 8, 8, 9, 10, 12, 12, 14, 14, 16, 18, 17, 18, 20, 20, 22, 24, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 33, 35, 36, 38, 38, 38, 40, 42, 42, 42, 44, 44, 46, 48, 48, 48, 50, 50, 52, 54, 55, 54, 56, 58, 58, 60, 60, 60, 62, 62, 64, 66, 65, 67, 68, 68, 70, 72, 73, 72, 74, 74

Offset: 2

Amarnath Murthy, Jul 03 2004

nonn

Conjecture: a(n)=n+1 only when n is prime or a power of a prime. [Verified for n=2..5000. - Amiram Eldar, Apr 06 2021]

			a(6) = 8 as 36!/(6!)^8 is an integer which is not further divisible by 720.

Amiram Eldar, Table of n, a(n) for n = 2..5000

Cf. A034841, A057599, A096126.

f[n_] := Block[{k = n}, While[ IntegerQ[(n^2)!/n!^k], k++ ]; k - 1]; Table[ f[n], {n, 75}] (* Robert G. Wilson v, Jul 03 2004 *)

Edited by Don Reble and Robert G. Wilson v, Jul 04 2004

A120666 Triangle read by rows: T(n, k) = (n*k)!/(n!)^k.

Original entry on oeis.org

1, 1, 6, 1, 20, 1680, 1, 70, 34650, 63063000, 1, 252, 756756, 11732745024, 623360743125120, 1, 924, 17153136, 2308743493056, 1370874167589326400, 2670177736637149247308800, 1, 3432, 399072960, 472518347558400, 3177459078523411968000, 85722533226982363751829504000, 7363615666157189603982585462030336000

Offset: 1

Roger L. Bagula, Aug 11 2006

T(m,n) is the number of ways to distribute n*m different toys among m different kids so that each kid gets exactly n toys. For example, with n=3 and m=2, the 6 different toys, t1, t2, t3, t4, t5 and t6, can be distributed in exactly 20 ways among the 2 kids, k1 and k2, since there are C(6,3)=20 ways to choose the three toys for k1, with the other three toys going to k2. The proof for the general case is based on the identity C(n*m,n)*C(n*m-n,n)*...*C(n*m-n*(m-1),n) = (n*m)!/(n!)^m. - Dennis P. Walsh, Apr 12 2018

			Triangle begins:
  1;
  1,   6;
  1,  20,   1680;
  1,  70,  34650,    63063000;
  1, 252, 756756, 11732745024, 623360743125120;

Seiichi Manyama, Rows n = 1..26, flattened
Eric Weisstein's World of Mathematics, Macdonald's Constant-Term Conjecture

Cf. A000984, A006480, A034841, A089759, A187783.

[Factorial(n*k)/(Factorial(n))^k: k in [1..n], n in [1..10]]; // G. C. Greubel, Dec 26 2022

T:= (m, n)-> (n*m)!/(m!)^n:
seq(seq(T(m, n), n=1..m), m=1..7);  # Alois P. Heinz, Apr 12 2018

Table[(n*k)!/(n!)^k, {n,10}, {k,n}]//Flatten

def A120666(n,k): return gamma(n*k+1)/(factorial(n))^k
flatten([[A120666(n,k) for k in range(1,n+1)] for n in range(1,11)]) # G. C. Greubel, Dec 26 2022

T(n, k) = (k*n)!/(n!)^k.

Edited by N. J. A. Sloane, Jun 17 2007
Offset corrected by Alois P. Heinz, Apr 12 2018
New name using formula by Joerg Arndt, Apr 15 2018

A141906 Triangle t(n,m) = (n*m)!/(m!^n) read by rows, 0<=m<=n.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 1, 6, 90, 1680, 1, 24, 2520, 369600, 63063000, 1, 120, 113400, 168168000, 305540235000, 623360743125120, 1, 720, 7484400, 137225088000, 3246670537110000, 88832646059788350720, 2670177736637149247308800

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 14 2008

Row sums are in A221177.

			1;
1, 1;
1, 2, 6;
1, 6, 90, 1680;
1, 24, 2520, 369600, 63063000;
1, 120, 113400, 168168000, 305540235000, 623360743125120;
1, 720, 7484400, 137225088000, 3246670537110000, 88832646059788350720, 2670177736637149247308800;

Seiichi Manyama, Rows n = 0..26, flattened

Cf. A034841, A187783, A221177.

A141906 := proc(n,m)
        (n*m)!/m!^n ;
end proc:
seq(seq(A141906(n,m),m=0..n),n=0..5) ; # R. J. Mathar, Nov 08 2011

Clear[t, n, m]; t[n_, m_] = (n*m)!/m!^n; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

A268751 Number of sequences with n copies each of 1,2,...,n avoiding the pattern 12...n.

Original entry on oeis.org

0, 0, 1, 374, 16140983, 173996758190594, 791857392420720220446647, 2285085934263252199073238394141449534, 5841526335200139692050292842849347521755651331941759, 17585875137049122003330684747231440185032966840579881629527695901745706

Offset: 0

Alois P. Heinz, Feb 12 2016

nonn

			a(2) = 1: 2211.
a(3) = 374: 111333222, 113133222, 113313222, ..., 333221121, 333221211, 333222111.

Alois P. Heinz, Table of n, a(n) for n = 0..27

Main diagonal of A269129.

Cf. A034841, A268485.

a(n) = A034841(n) - A268485(n).

A340590 Number of n*(n+1)-step n-dimensional nonnegative closed lattice walks starting at the origin and using steps that increment all components or decrement one component by 1.

Original entry on oeis.org

1, 1, 16, 24444, 8204167296, 1052109889288796160, 78607706455594117933558272000, 4825997038234002956322487606996722432307200, 325844502690869718672482402463320899403011435565608069632000, 31176247959648026790291638390172796940342899651173947284143811081979726010777600

Offset: 0

Alois P. Heinz, Jan 12 2021

			a(2) = 16:
  [(0,0),(1,1),(0,1),(0,0),(1,1),(0,1),(0,0)],
  [(0,0),(1,1),(0,1),(0,0),(1,1),(1,0),(0,0)],
  [(0,0),(1,1),(0,1),(1,2),(0,2),(0,1),(0,0)],
  [(0,0),(1,1),(0,1),(1,2),(1,1),(0,1),(0,0)],
  [(0,0),(1,1),(0,1),(1,2),(1,1),(1,0),(0,0)],
  [(0,0),(1,1),(1,0),(0,0),(1,1),(0,1),(0,0)],
  [(0,0),(1,1),(1,0),(0,0),(1,1),(1,0),(0,0)],
  [(0,0),(1,1),(1,0),(2,1),(1,1),(0,1),(0,0)],
  [(0,0),(1,1),(1,0),(2,1),(1,1),(1,0),(0,0)],
  [(0,0),(1,1),(1,0),(2,1),(2,0),(1,0),(0,0)],
  [(0,0),(1,1),(2,2),(1,2),(0,2),(0,1),(0,0)],
  [(0,0),(1,1),(2,2),(1,2),(1,1),(0,1),(0,0)],
  [(0,0),(1,1),(2,2),(1,2),(1,1),(1,0),(0,0)],
  [(0,0),(1,1),(2,2),(2,1),(1,1),(0,1),(0,0)],
  [(0,0),(1,1),(2,2),(2,1),(1,1),(1,0),(0,0)],
  [(0,0),(1,1),(2,2),(2,1),(2,0),(1,0),(0,0)].

Alois P. Heinz, Table of n, a(n) for n = 0..12

Main diagonal of A340591.

Cf. A002378, A034841.

b:= proc(n, l) option remember; `if`(n=0, 1, (k-> add(
     `if`(l[i]>0, b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..k)+
     `if`(add(i, i=l)+k x+1, l)), 0))(nops(l)))
    end:
a:= n-> b(n*(n+1), [0$n]):
seq(a(n), n=0..9);

b[n_, l_] := b[n, l] = If[n == 0, 1, Function[k, Sum[
    If[l[[i]]>0, b[n-1, Sort[ReplacePart[l, i -> l[[i]]-1]]], 0], {i, 1, k}] +
    If[Sum[i, {i, l}] + k < n, b[n - 1, Map[#+1&, l]], 0]][Length[l]]];
a[n_] := b[n(n+1), Table[0, {n}]];
a /@ Range[0, 9] (* Jean-François Alcover, Jan 26 2021, after Alois P. Heinz *)

a(n) = A340591(n,n).

A345466 a(n) = Product_{k=1..n} binomial(n, floor(n/k)).

Original entry on oeis.org

1, 1, 2, 9, 96, 1250, 64800, 1764735, 224788480, 22499086176, 6123600000000, 408514437465750, 1308805762115174400, 133962125607455951520, 99335199198879310098432, 113040832521732593994140625, 425230288403106927476736000000, 72623663171934137824096600064000

Offset: 0

Vaclav Kotesovec, Jun 20 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..160

Cf. A010786, A034841, A051054, A092143, A272093.

[n eq 0 select 1 else (&*[Binomial(n,Floor(n/j)): j in [1..n]]): n in [0..30]]; // G. C. Greubel, Feb 05 2024

Table[Product[Binomial[n, Floor[n/k]], {k, 1, n}], {n, 0, 20}]
Table[Product[((n + 1)/k - 1)^Floor[n/k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 24 2021 *)

[product(binomial(n,(n//j)) for j in range(1,n+1)) for n in range(31)] # G. C. Greubel, Feb 05 2024

log(a(n)) ~ n * log(n)^2 / 2. - Vaclav Kotesovec, Jun 21 2021

a(n) = Product_{k=1..n} ((n+1)/k - 1)^floor(n/k). - Vaclav Kotesovec, Jun 24 2021

A375693 Number of multiset permutations of {{1}^n, {2}^n, ..., {n}^n} with no fixed n-tuple {j}^n.

Original entry on oeis.org

1, 0, 5, 1622, 62924817, 623302086965044, 2670169511426774520697375, 7363615066099523741730150062678073534, 18165723898797467057177720588121375861340650728031233, 53130688704554689391452744667655289291011354800478739192999936981375688

Offset: 0

Alois P. Heinz, Aug 24 2024

nonn

			a(2) = 5: 1212, 1221, 2112, 2121, 2211.

Alois P. Heinz, Table of n, a(n) for n = 0..26

Main diagonal of A375694.

Cf. A034841, A059841, A306241, A375778.

a:= n-> add((-1)^(n-j)*binomial(n, j)*(n*j)!/n!^j, j=0..n):
seq(a(n), n=0..10);

a(n) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(n*j)!/n!^j.

a(n) mod 2 = 1 - (n mod 2) = A059841(n).

A378382 Number of maximal chains in the poset of all binary words of length <= n, ordered by B covers A iff A_i <= B_{i+k} for all i in A and some k >= 0.

Original entry on oeis.org

1, 1, 2, 5, 16, 57, 226, 961, 4376, 21041, 106534, 563961, 3112924, 17839993, 105907946, 649432673, 4105783696, 26706965985, 178466243662, 1223248786921, 8589272300516, 61708802126441, 453143009601682, 3397715981566545, 25990997059282456, 202666687407866257

Offset: 0

John Tyler Rascoe, Nov 26 2024

nonn

			a(3) = 5:
 () < (0) < (0,0) < (0,0,0),
 () < (0) < (0,0) < (0,1),
 () < (0) < (0,0) < (1,0),
 () < (0) < (1) < (0,1),
 () < (0) < (1) < (1,0).

Cf. A034841, A143672, A282698, A317145, column k=2 of A378588, A378608.

def mchains(n, k): return # See A378588
def A378382_list(max_n): return mchains(max_n,2)

A229051 G.f.: Sum_{n>=0} (n^2)!/n!^n * (2*x)^n / (1-x)^(n^2+1).

Original entry on oeis.org

1, 3, 29, 13567, 1009142769, 19947560933879891, 170891375663413844489045533, 942542805274443250197129297402029958879, 4650425326497533656923054675764068523027405525255181377, 27202912617670436035808496756146798219927348043651854025145948950565355779

Offset: 0

Paul D. Hanna, Sep 22 2013

nonn

			G.f.: A(x) = 1 + 3*x + 29*x^2 + 13567*x^3 + 1009142769*x^4 +...
where
A(x) = 1/(1-x) + (2*x)/(1-x)^2 + (4!/2!^2)*(2*x)^2/(1-x)^5 + (9!/3!^3)*(2*x)^3/(1-x)^10 + (16!/4!^4)*(2*x)^4/(1-x)^17 + (25!/5!^5)*(2*x)^5/(1-x)^26 +...
Equivalently,
A(x) = 1/(1-x) + (2*x)/(1-x)^2 + 6*(2*x)^2/(1-x)^5 + 1680*(2*x)^3/(1-x)^10 + 63063000*(2*x)^4/(1-x)^17 + 623360743125120*(2*x)^5/(1-x)^26 +...+ A034841(n)*(2*x)^n/(1-x)^(n^2+1) +...
Illustrate formula a(n) = Sum_{k=0..n} 2^k * Product_{j=0..k-1} C(n+j*k,k) for initial terms:
a(0) = 1;
a(1) = 1 + 2*C(1,1);
a(2) = 1 + 2*C(2,1) + 4*C(2,2)*C(4,2);
a(3) = 1 + 2*C(3,1) + 4*C(3,2)*C(5,2) + 8*C(3,3)*C(6,3)*C(9,3);
a(4) = 1 + 2*C(4,1) + 4*C(4,2)*C(6,2) + 8*C(4,3)*C(7,3)*C(10,3) + 16*C(4,4)*C(8,4)*C(12,4)*C(16,4);
a(5) = 1 + 2*C(5,1) + 4*C(5,2)*C(7,2) + 8*C(5,3)*C(8,3)*C(11,3) + 16*C(5,4)*C(9,4)*C(13,4)*C(17,4) + 32*C(5,5)*C(10,5)*C(15,5)*C(20,5)*C(25,5); ...
which numerically equals:
a(0) = 1;
a(1) = 1 + 2*1 = 3;
a(2) = 1 + 2*2 + 4*1*6 = 29;
a(3) = 1 + 2*3 + 4*3*10 + 8*1*20*84 = 13567;
a(4) = 1 + 2*4 + 4*6*15 + 8*4*35*120 + 16*1*70*495*1820 = 1009142769;
a(5) = 1 + 2*5 + 4*10*21 + 8*10*56*165 + 16*5*126*715*2380 + 32*1*252*3003*15504*53130 = 19947560933879891; ...

Alois P. Heinz, Table of n, a(n) for n = 0..26

Cf. A229050, A034841; A001850, A081798, A082488, A082489, A229049.

with(combinat):
a:= n-> add(2^k*multinomial(n+(k-1)*k, n-k, k$k), k=0..n):
seq(a(n), n=0..10);  # Alois P. Heinz, Sep 23 2013

Table[Sum[2^k*Product[Binomial[n+j*k,k],{j,0,k-1}],{k,0,n}],{n,0,10}] (* Vaclav Kotesovec, Sep 23 2013 *)

{a(n)=polcoeff(sum(m=0, n, (m^2)!/m!^m*(2*x)^m/(1-x+x*O(x^n))^(m^2+1)), n)}
for(n=0,15,print1(a(n),", "))

{a(n)=sum(k=0,n,2^k*prod(j=0,k-1,binomial(n+j*k,k)))}
for(n=0,15,print1(a(n),", "))

a(n) = Sum_{k=0..n} 2^k * Product_{j=0..k-1} binomial(n+j*k,k).

a(n) ~ exp(-1/12) * n^(n^2-n/2+1) * 2^n / (2*Pi)^((n-1)/2). - Vaclav Kotesovec, Sep 23 2013

A306241 a(n) = Sum_{k=0..n} (k*n)!/n!^k.

Original entry on oeis.org

1, 2, 8, 1702, 63097722, 623372476627154, 2670179107513625597282318, 7363615751879726008424500256018442794, 18165723935734974232438957032838329596079311234990642

Offset: 0

Seiichi Manyama, Jan 31 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A034841, A120666, A221177.

Table[Sum[(k*n)!/n!^k,{k,0,n}],{n,0,10}] (* Vaclav Kotesovec, Feb 08 2019 *)

PARI
```
{a(n) = sum(k=0, n, (k*n)!/n!^k)}
```

a(n) equals (row sums of A120666) + 1.
From Vaclav Kotesovec, Feb 08 2019: (Start)
a(n) ~ A034841(n).
a(n) ~ n^(n^2 - n/2 + 1) / (exp(1/12) * (2*Pi)^((n-1)/2)). (End)

cp's OEIS Frontend

A096127 a(n) is the largest k such that (n^2)!/(n!)^k is an integer.

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A120666 Triangle read by rows: T(n, k) = (n*k)!/(n!)^k.

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A141906 Triangle t(n,m) = (n*m)!/(m!^n) read by rows, 0<=m<=n.

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Maple

Mathematica

A268751 Number of sequences with n copies each of 1,2,...,n avoiding the pattern 12...n.

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Author

Keywords

Examples

Links

Crossrefs

Formula

A340590 Number of n*(n+1)-step n-dimensional nonnegative closed lattice walks starting at the origin and using steps that increment all components or decrement one component by 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A345466 a(n) = Product_{k=1..n} binomial(n, floor(n/k)).

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Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A375693 Number of multiset permutations of {{1}^n, {2}^n, ..., {n}^n} with no fixed n-tuple {j}^n.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A378382 Number of maximal chains in the poset of all binary words of length <= n, ordered by B covers A iff A_i <= B_{i+k} for all i in A and some k >= 0.

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