A000680 -id:A000680 - cp's OEIS frontend

A374980 Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed pair (j,j).

1, 0, 5, 74, 2193, 101644, 6840085, 630985830, 76484389121, 11792973495032, 2254432154097861, 523368281765512930, 145044815855963403985, 47302856057098946329284, 17933275902554972391519893, 7820842217155394547769452734, 3887745712142302082441578104705

Offset: 0

Alois P. Heinz, Aug 05 2024

nonn

Inverse binomial transform of A000680.

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

Alois P. Heinz, Table of n, a(n) for n = 0..238

Cf. A000166, A000459, A000680, A059841, A116218, A375222, A375223.

Column k=2 of A375694.

a:= proc(n) option remember; `if`(n<3, [1, 0, 5][n+1],
     (n-1)*((2*n+1)*a(n-1)+(4*n-3)*a(n-2)+2*(n-2)*a(n-3)))
    end:
seq(a(n), n=0..16);

a(n) = Sum_{j=0..n} (-1)^j*binomial(n,j)*A000680(n-j).
a(n) = A116218(n)/2^n.
a(n) mod 2 = 1 - (n mod 2) = A059841(n).

A067630 Denominators in power series for cos(x)*cosh(x).

Original entry on oeis.org

1, 6, 2520, 7484400, 81729648000, 2375880867360000, 151476660579404160000, 18608907752179801056000000, 4015057936610313875842560000000, 1419041926536183233139035980800000000, 778117449996850714059458989711872000000000

Offset: 0

Benoit Cloitre, Feb 02 2002

nonn

Cf. A000680, A060706.

f:= gfun:-rectoproc({a(n) - (64*n^4-96*n^3+44*n^2-6*n)*a(n-1), a(0)=1}, a(n), remember): map(f, [$0..20]); # Georg Fischer, Aug 17 2021

a[n_] := (4*n)!/4^n; Array[a, 10, 0] (* Amiram Eldar, Jan 18 2021 *)

my(x='x+O('x^50), v=apply(denominator, Vec(cos(x)*cosh(x)))); vector(#v\4, k, v[4*k-3]) \\ Michel Marcus, Jan 18 2021

cos(x)*cosh(x) = Sum_{n>=0} (-1)^n*x^(4*n)/a(n).
a(n) = (4*n)! / 4^n = A000680(2*n).
E.g.f.: 1/(1-x^4/4). - Mohammad K. Azarian, Mar 20 2012
a(n) = n!*A060706(n). - Bruno Berselli, Mar 21 2012
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=0} 1/a(n) = (cos(sqrt(2)) + cosh(sqrt(2)))/2.
Sum_{n>=0} (-1)^n/a(n) = cos(1)*cosh(1). (End)
D-finite with recurrence: a(n) - (64*n^4 - 96*n^3 + 44*n^2 - 6*n)*a(n-1) = 0. - Georg Fischer, Aug 17 2021

A210277 a(n) = (3*n)!/3^n.

Original entry on oeis.org

1, 2, 80, 13440, 5913600, 5381376000, 8782405632000, 23361198981120000, 94566133475573760000, 553211880832106496000000, 4492080472356704747520000000, 49017582114356362204938240000000, 699971072593008852286518067200000000

Offset: 0

Mohammad K. Azarian, Mar 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..100
D. Bevan, D. Levin, P. Nugent, J. Pantone and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015-2016.

Cf. A210278, A000680, A067630, A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101.

[Factorial(3*n)/3^n: n in [0..15]]; // Vincenzo Librandi, Feb 15 2013

Table[(3 n)!/3^n, {n, 0, 15}] (* Vincenzo Librandi, Feb 15 2013 *)

E.g.f.: 1/(1-x^3/3).
a(n) = Product_{i=1..n} (2*binomial(3i,3)). - James Mahoney, Apr 04 2012
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=0} 1/a(n) = exp(3^(1/3))/3 + (2/3)*exp(-3^(1/3)/2)*cos(3^(5/6)/2).
Sum_{n>=0} (-1)^n/a(n) = exp(-3^(1/3))/3 + (2/3)*exp(3^(1/3)/2)*cos(3^(5/6)/2). (End)

A210278 (5n)!/5^n.

Original entry on oeis.org

1, 24, 145152, 10461394944, 3892643213082624, 4963587213865915514880, 16976183027980227752723742720, 132264293969742655099733137120296960, 2088743125114618199924764850166056689336320, 61246577083125859615725138685776750112964471685120

Offset: 0

Mohammad K. Azarian, Mar 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..90

Cf. A210277, A000680, A067630, A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101.

[Factorial(5*n)/5^n: n in [0..10]]; // Vincenzo Librandi, Feb 15 2013

Table[(5 n)!/5^n, {n, 0, 10}] (* Vincenzo Librandi, Feb 15 2013 *)
With[{nn=100},Take[CoefficientList[Series[1/(1-x^5/5),{x,0,nn}],x] Range[0,nn]!,{1,-1,5}]] (* Harvey P. Dale, May 27 2025 *)

E.g.f.: 1/(1-x^5/5).

A248686 Triangular array of multinomial coefficients: T(n,k) = n!/(n(1)!n(2)! ... *n(k)!), where n(i) = floor((n + i - 1)/k) for i = 1 .. k.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 6, 12, 24, 1, 10, 30, 60, 120, 1, 20, 90, 180, 360, 720, 1, 35, 210, 630, 1260, 2520, 5040, 1, 70, 560, 2520, 5040, 10080, 20160, 40320, 1, 126, 1680, 7560, 22680, 45360, 90720, 181440, 362880, 1, 252, 4200, 25200, 113400, 226800, 453600, 907200, 1814400, 3628800

Offset: 1

Clark Kimberling, Oct 11 2014

T(n,k) is the number of permutations p of [n] such that p(i)Alois P. Heinz, Feb 09 2023

			First seven rows:
  1
  1    2
  1    3     6
  1    6    12   24
  1   10    30   60    120
  1   20    90  180    360    720
  1   35   210  630   1260   2520   5040
  ...
Writing floor as [ ], the numbers comprising row 4 are
T(4,1) = 4!/[4/1]! = 24/24 = 1
T(4,2) = 4!/([4/2]![5/2]!) = 24/(2*2) = 6
T(4,3) = 4!/([4/3]![5/3]![6/3]!) = 24/(1*1*2) = 12
T(4,4) = 4!/([4/4]![5/4]![6/4]![7/4]!) = 24/(1*1*1*1) = 24.

Clark Kimberling, Table of n, a(n) for n = 1..5000

Main diagonal is A000142.
T(2n,n) gives A000680.
Row sums give A248687.
Cf. A333706.

T:= (n, k)-> combinat[multinomial](n, floor((n+i)/k)$i=0..k-1):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Feb 09 2023

f[n_, k_] := f[n, k] = n!/Product[Floor[(n + i)/k]!, {i, 0, k - 1}]
t = Table[f[n, k], {n, 0, 10}, {k, 1, n}];
u = Flatten[t]  (* A248686 sequence *)
TableForm[t]    (* A248686 array *)
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 22}] (* A248687 *)

A267480 Number T(n,k) of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 5, 0, 1, 42, 47, 0, 1, 351, 1527, 641, 0, 1, 3113, 43910, 54987, 11389, 0, 1, 29003, 1302660, 3844840, 2059147, 248749, 0, 1, 280220, 40970298, 265777225, 285588543, 82025038, 6439075, 0, 1, 2782475, 1364750889, 19104601915, 37783672691, 19773928713, 3507289363, 192621953

Offset: 0

Alois P. Heinz, Jan 15 2016

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,      5;
  0, 1,     42,       47;
  0, 1,    351,     1527,       641;
  0, 1,   3113,    43910,     54987,     11389;
  0, 1,  29003,  1302660,   3844840,   2059147,   248749;
  0, 1, 280220, 40970298, 265777225, 285588543, 82025038, 6439075;

Alois P. Heinz, Rows n = 0..18, flattened
Ferenc Balogh, A generalization of Gessel's generating function to enumerate words with double or triple occurrences in each letter and without increasing subsequences of a given length, arXiv:1505.01389, 2015
Shalosh B. Ekhad and Doron Zeilberger, The Generating Functions Enumerating 12..d-Avoiding Words with r occurrences of each of 1,2, ..., n are D-finite for all d and all r, 2014

Main diagonal gives A006902.
Row sums give A000680.
Cf. A267479, A267532.

T(n,0) = A267479(n,0), T(n,k) = A267479(n,k) - A267479(n,k-1) for k>0.

Sum_{k=0..n-1} T(n,k) = A267532(n).

A267532 Number of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length < n.

Original entry on oeis.org

0, 0, 1, 43, 1879, 102011, 7235651, 674641325, 81537026047, 12498099730471, 2375632826877259, 548818073236649129, 151476182218777630655, 49229890784448694885163, 18608906461974462064310179, 8094874797394331233877338741, 4015057931973886657462193434111

Offset: 0

Alois P. Heinz, Jan 16 2016

nonn

Or number of words on {1,1,2,2,...,n,n} avoiding the pattern 12...n.

			a(2) = 1: 2211.
a(3) = 43: 113322, 131322, 133122, 133212, 133221, 211332, 213132, 213312, 213321, 221133, 221313, 221331, 223113, 223131, 223311, 231132, 231312, 231321, 232113, 232131, 232311, 233112, 233121, 233211, 311322, 313122, 313212, 313221, 321132, 321312, 321321, 322113, 322131, 322311, 323112, 323121, 323211, 331122, 331212, 331221, 332112, 332121, 332211.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000079, A000142, A000680, A006902, A010050, A267479, A267480, A269042.

Column k=2 of A269129.

b:= proc(n) option remember; `if`(n<3, [1$2, 5][n+1], (
      (n^3+n^2-7*n+4)*b(n-1)-2*(2*n-3)*(n-1)^3*b(n-2))/(n-2))
    end:
a:= n-> (2*n)!/(2^n)-b(n):
seq(a(n), n=0..20);

a(n) = (2*n)! * ( 1/(2^n) - Sum_{k=0..n} (-1)^k * C(n,k) / (n+k)! ).
a(n) = A000680(n) - A006902(n).
a(n) = A267479(n,n-1) for n>0.
a(n) = Sum_{k=0..n-1} A267480(n,k).

A375223 a(n) is the number of permutations of the multiset 1,1, 2,2, ..., n,n such that at least one pair k,k stays at its initial locations 2k-1, 2k.

Original entry on oeis.org

1, 1, 16, 327, 11756, 644315, 50094570, 5245258879, 711662648968, 121448713262139, 25460198594647070, 6431844723440756015, 1927058631207405670716, 675631849624828664480107, 274032655042818911590547266, 127312224468011793400981895295, 67167619760422081463964260973200

Offset: 1

Hugo Pfoertner, Aug 05 2024

nonn

			a(3) = 16: The 15 permutations with one stable pair (see A375222) and the starting configuration [1, 1, 2, 2, 3, 3].

Alois P. Heinz, Table of n, a(n) for n = 1..239

Cf. A000680 (all permutations of this multiset), A375222 (exactly one stable pair), A374980.

a375223(n) = {my (p=vector(2*n,i,1+(i-1)\2), m=0); forperm (p, q, for (j=1, n, if (q[2*j-1]==j && q[2*j]==j, m++; break))); m}

a(n) = Sum_{j=1..n} binomial(n,j) * A374980(n-j). - Alois P. Heinz, Aug 05 2024

a(8) onwards from Alois P. Heinz, Aug 05 2024

A059117 Square array of lambda(k,n), where lambda is defined in A055203. Number of ways of placing n identifiable positive intervals with a total of exactly k starting and/or finishing points.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 6, 24, 1, 0, 0, 0, 0, 0, 114, 78, 1, 0, 0, 0, 0, 0, 180, 978, 240, 1, 0, 0, 0, 0, 0, 90, 4320, 6810, 726, 1, 0, 0, 0, 0, 0, 0, 8460, 63540, 43746, 2184, 1, 0, 0, 0, 0, 0, 0, 7560, 271170, 774000, 271194, 6558, 1

Offset: 0

Henry Bottomley, Jan 05 2001

			Rows are: 1,0,0,0,0,0,....; 0,0,1,0,0,0,....; 0,0,1,6,6,0,....; 0,0,1,24,114,180,.... etc.

Sum of rows gives A055203. Columns include A000007, A057427, A058809, A059116. Final positive number in each row is A000680.

A[ n_, k_] := If[n < 1 || k < 1, Boole[n == 0 && k == 0], n! k! Coefficient[ Normal[ Series[ Sum[ Exp[-x z] (x z)^m/m! Exp[y z m (m - 1)/2], {m, 0, n}], {z, 0, n + k}]], x^n y^k z^(n + k)]]; (* Michael Somos, Jul 17 2019 *)

lambda(k, n) = (lambda(k - 2, n - 1) + 2*lambda(k - 2, n - 1) + lambda(k - 2, n - 1))*k*(k - 1)/2 starting with lambda(k, 0) = 1 if k = 0 but = 0 otherwise. lambda(k, n) = sum_{j=0..k} (-1)^(k + j) * C(k, j) * ((j - 1)*j/2)^n.

A059515 Square array T(k,n) by antidiagonals, where T(k,n) is number of ways of placing n identifiable nonnegative intervals with a total of exactly k starting and/or finishing points.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 7, 1, 0, 0, 0, 12, 25, 1, 0, 0, 0, 6, 138, 79, 1, 0, 0, 0, 0, 294, 1056, 241, 1, 0, 0, 0, 0, 270, 5298, 7050, 727, 1, 0, 0, 0, 0, 90, 12780, 70350, 44472, 2185, 1, 0, 0, 0, 0, 0, 16020, 334710, 817746, 273378, 6559, 1, 0, 0, 0, 0, 0

Offset: 0

Henry Bottomley, Jan 19 2001

See A300729 for a triangular version of this array. - Peter Bala, Jun 13 2019

			Rows are: 1,0,0,0,0,..., 0,1,1,0,0,..., 0,1,7,12,6,..., 0,1,25,138,294,..., etc. T(1,1)=1 since if a is starting point of interval and A is end point then only possibility is aA (zero length). T(2,1)=1 since possibility is a-A (positive length). T(3,2)=12 since possibilities are: aA-b-B, b-aA-B, b-B-aA, bB-a-A, a-bB-A, a-A-bB, ab-A-B, ab-B-A, a-b-AB, b-a-AB, a-bA-B, b-a-AB.

IBM Ponder This, Jan 01 2001

Sum of rows gives A059516. Columns include A000007, A057427, A058481, A059117. Final positive number in each row is A000680.

Cf. A300729.

T(k, n) = T(k - 2, n - 1) * k * (k - 1)/2 + T(k - 1, n - 1) * k^2 + T(k, n - 1) * k * (k + 1)/2 with T(0, 0) = 1 = lambda(k, n) + lambda(k + 1, n) where lambda is A059117(k, n).

cp's OEIS Frontend

A374980 Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed pair (j,j).

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A067630 Denominators in power series for cos(x)*cosh(x).

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A210277 a(n) = (3*n)!/3^n.

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Magma

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A210278 (5n)!/5^n.

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Magma

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A248686 Triangular array of multinomial coefficients: T(n,k) = n!/(n(1)!*n(2)!* ... *n(k)!), where n(i) = floor((n + i - 1)/k) for i = 1 .. k.

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A267480 Number T(n,k) of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A267532 Number of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length < n.

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A375223 a(n) is the number of permutations of the multiset 1,1, 2,2, ..., n,n such that at least one pair k,k stays at its initial locations 2k-1, 2k.

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A248686 Triangular array of multinomial coefficients: T(n,k) = n!/(n(1)!n(2)! ... *n(k)!), where n(i) = floor((n + i - 1)/k) for i = 1 .. k.