A089759 -id:A089759 - cp's OEIS frontend

A000680 a(n) = (2n)!/2^n.

1, 1, 6, 90, 2520, 113400, 7484400, 681080400, 81729648000, 12504636144000, 2375880867360000, 548828480360160000, 151476660579404160000, 49229914688306352000000, 18608907752179801056000000, 8094874872198213459360000000, 4015057936610313875842560000000

Offset: 0

N. J. A. Sloane

Denominators in the expansion of cos(sqrt(2)*x) = 1 - (sqrt(2)*x)^2/2! + (sqrt(2)*x)^4/4! - (sqrt(2)*x)^6/6! + ... = 1 - x^2 + x^4/6 - x^6/90 + ... By Stirling's formula in A000142: a(n) ~ 2^(n+1) * (n/e)^(2n) * sqrt(Pi*n) - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 20 2001
a(n) is also the constant term in the product: Product_{1<=i, j<=n, i!=j} (1 - x_i/x_j)^2. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 12 2002
a(n) is also the number of lattice paths in the n-dimensional lattice [0..2]^n. - T. D. Noe, Jun 06 2002
Representation as the n-th moment of a positive function on the positive half-axis: a(n) = Integral_{x>=0} (x^n*exp(-sqrt(2*x))/sqrt(2*x)), n=0,1,... - Karol A. Penson, Mar 10 2003
Number of permutations of [2n] with no increasing runs of odd length. Example: a(2) = 6 because we have 1234, 13/24, 14/23, 23/14, 24/13 and 34/12 (runs separated by slashes). - Emeric Deutsch, Aug 29 2004
This is also the number of ways of arranging the elements of n distinct pairs, assuming the order of elements is significant and the pairs are distinguishable. When the pairs are not distinguishable, see A001147 and A132101. For example, there are 6 ways of arranging 2 pairs [1,1], [2,2]: {[1122], [1212], [1221], [2211], [2121], [2112]}. - Ross Drewe, Mar 16 2008
n married couples are seated in a row so that every wife is to the left of her husband. The recurrence a(n+1) = a(n)*((2*n + 1) + binomial(2*n+1, 2)) conditions on whether the (n+1)st couple is seated together or separated by at least one other person. - Geoffrey Critzer, Jun 10 2009
a(n) is the number of functions f:[2n]->[n] such that the preimage of {y} has cardinality 2 for every y in [n]. Note that [k] denotes the set {1,2,...,k} and [0] denotes the empty set. - Dennis P. Walsh, Nov 17 2009
a(n) is also the number of n X 2n (0,1)-matrices with row sum 2 and column sum 1. - Shanzhen Gao, Feb 12 2010
Number of ways that 2n people of different heights can be arranged (for a photograph) in two rows of equal length so that every person in the front row is shorter than the person immediately behind them in the back row.
a(n) is the number of functions f:[n]->[n^2] such that, if floor((f(x))^.5) = floor((f(y))^.5), then x = y. For example, with n = 4, the range of f consists of one element from each of the four sets {1,2,3}, {4,5,6,7,8}, {9,10,11,12,13,14,15}, and {16}. Hence there are 1*3*5*7 = 105 ways to choose the range for f, and there are 4! ways to injectively map {1,2,3,4} to the four elements of the range. Thus there are 105*24 = 2520 such functions. Note also that a(n) = n!*(product of the first n odd numbers). - Dennis P. Walsh, Nov 28 2012
a(n) is also the 2*n th difference of n-powers of A000217 (triangular numbers). For example a(2) is the 4th difference of the squares of triangular numbers. - Enric Reverter i Bigas, Jun 24 2013
a(n) is the multinomial coefficient (2*n) over (2, 2, 2, ..., 2) where there are n 2's in the last parenthesis. It is therefore also the number of words of length 2n obtained with n letters, each letter appearing twice. - Robert FERREOL, Jan 14 2018
Number of ways to put socks and shoes on an n-legged animal, if a sock must be put on before a shoe. - Daniel Bishop, Jan 29 2018

			For n = 2, a(2) = 6 since there are 6 functions f:[4]->[2] with size 2 preimages for both {1} and {2}. In this case, there are binomial(4, 2) = 6 ways to choose the 2 elements of [4] f maps to {1} and the 2 elements of [4] that f maps to {2}. - _Dennis P. Walsh_, Nov 17 2009

G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1998.
H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 283.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
Shanzhen Gao and Kenneth Matheis, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 45-53.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. B. Tompkins, Methods of successive restrictions in computational problems involving discrete variables. 1963, Proc. Sympos. Appl. Math., Vol. XV pp. 95-106; Amer. Math. Soc., Providence, R.I.

T. D. Noe, Table of n, a(n) for n = 0..100
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.
R. Florez and L. Junes, A relation between triangular numbers and prime numbers, Integers 12(1) (2012), 83-96.
M. Ghebleh, Antichains of (0, 1)-matrices through inversions, Linear Algebra and its Applications 458 (2014), 503-511.
S. A. Joffe, Calculation of the first thirty-two Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 47 (1914), 103-126. [Accessible only in the USA through the Hathi Trust Digital Library.]
Peter D. Loly and Ian D. Cameron, Frierson's 1907 Parameterization of Compound Magic Squares Extended to Orders 3^L, L = 1, 2, 3, ..., with Information Entropy, arXiv:2008.11020 [math.HO], 2020.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Dennis Walsh, Counting integer functions with size-2 preimage constraints, preprint.
Eric Weisstein's World of Mathematics, Lattice Path.
Index to divisibility sequences
Index entries for related partition-counting sequences

Cf. A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101.
A diagonal of the triangle in A241171.
Main diagonal of A267479, row sums of A267480.
Row n=2 of A089759.
Column n=2 of A187783.
Even bisection of column k=0 of A097591.

A000680 := n->(2*n)!/(2^n);
a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]*(2*n-1)*n od: seq(a[n], n=0..16); # Zerinvary Lajos, Mar 08 2008
seq(product(binomial(2*n-2*k,2),k=0..n-1),n=0..16); # Dennis P. Walsh, Nov 17 2009

Table[Product[Binomial[2 i, 2], {i, 1, n}], {n, 0, 16}]
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[ polygorial[6, #] &, 17, 0] (* Robert G. Wilson v, Dec 26 2016 *)
Table[(2n)!/2^n,{n,0,20}] (* Harvey P. Dale, Sep 21 2020 *)

PARI
```
a(n) = (2*n)! / 2^n
```

E.g.f.: 1/(1 - x^2/2) (with interpolating zeros). - Paul Barry, May 26 2003
a(n) = polygorial(n, 6) = (A000142(n)/A000079(n))*A001813(n) = (n!/2^n)*Product_{i=0..n-1} (4*i + 2) = (n!/2^n)*4^n*Pochhammer(1/2, n) = gamma(2*n+1)/2^n. - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
a(n) = A087127(n,2*n) = Sum_{i=0..2*n} (-1)^(2*n-i)*binomial(2*n, i)*binomial(i+2, 2)^n. Let T(n,k,j) = ((n - k + j)*(2*n - 2*k + 1))^n*binomial(2*n, 2*k-j+1) then a(n) = Sum{k=0..n} (T(n,k,1) - T(n,k,0)). For example a(12) = A087127(12,24) = Sum_{k=0..12} (T(12,k,1) - T(12,k,0)) = 24!/2^12. - André F. Labossière, Mar 29 2004 [Corrected by Jianing Song, Jan 08 2019]
For even n, a(n) = binomial(2n, n)*(a(n/2))^2. For odd n, a(n) = binomial(2n, n+1)*a((n+1)/2)*a((n-1)/2). For positive n, a(n) = binomial(2n, 2)*a(n-1) with a(0) = 1. - Dennis P. Walsh, Nov 17 2009
a(n) = Product_{i=1..n} binomial(2i, 2).
a(n) = a(n-1)*binomial(2n, 2).
From Peter Bala, Feb 21 2011: (Start)
a(n) = Product_{k = 0..n-1} (T(n) - T(k)), where T(n) = n*(n + 1)/2 is the n-th triangular number.
Compare with n! = Product_{k = 0..n-1} (n - k).
Thus we may view a(n) as a generalized factorial function associated with the triangular numbers A000217. Cf. A010050. The corresponding generalized binomial coefficients a(n)/(a(k)*a(n-k)) are triangle A086645. Also cf. A186432.
a(n) = n*(n + n-1)*(n + n-1 + n-2)*...*(n + n-1 + n-2 + ... + 1).
For example, a(5) = 5*(5+4)*(5+4+3)*(5+4+3+2)*(5+4+3+2+1) = 113400. (End).
G.f.: 1/U(0) where U(k)= x*(2*k - 1)*k + 1 - x*(2*k + 1)*(k + 1)/U(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Oct 28 2012
a(n) = n!*(product of the first n odd integers). - Dennis P. Walsh, Nov 28 2012
a(0) = 1, a(n) = a(n-1)*T(2*n-1), where T(n) is the n-th triangular number. For example: a(4) = a(3)*T(7) = 90*28 = 2520. - Enric Reverter i Bigas, Jun 24 2013
E.g.f.: 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = cosh(sqrt(2)).
Sum_{n>=0} (-1)^n/a(n) = cos(sqrt(2)). (End)
D-finite with recurrence a(n) -n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jan 28 2022
a(n) = n *A007019(n-1), n>0. - R. J. Mathar, Jan 28 2022

A014606 a(n) = (3n)!/(6^n).

Original entry on oeis.org

1, 1, 20, 1680, 369600, 168168000, 137225088000, 182509367040000, 369398958888960000, 1080491954750208000000, 4386797336285844480000000, 23934366266775567482880000000, 170891375144777551827763200000000, 1561776277448122046153927884800000000

Offset: 0

BjornE (mdeans(AT)algonet.se)

a(n) is also the constant term in the product : product 1 <= i,j <= n, i different from j (1 - x_i/x_j)^3. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 14 2002

a(n) is also the number of n by 3n (0,1)-matrices with row sum 3 and column sum 1. In general, the number of n by s*n (0,1)-matrices with row sum s and column sum 1 is (s*n)!/(s!)^n. - Shanzhen Gao, Feb 12 2010

George E. Andrews, Richard Askey and Ranjan Roy, Special Functions, Cambridge University Press, 1998.
Shanzhen Gao and Kenneth Matheis, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer., Vol. 202 (2010), pp. 45-53.

Alois P. Heinz, Table of n, a(n) for n = 0..165
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.

Cf. A000680, A025035, A089759.

nn=36;Select[Range[0,nn]!CoefficientList[Series[1/(1-x^3/3!),{x,0,nn}],x],#>0&] (* Geoffrey Critzer, Jun 07 2014 *)

PARI
```
a(n)=(3*n)!/6^n;
```

E.g.f. with interpolated zeros: 1/(1 - x^3/3!). - Geoffrey Critzer, Jun 07 2014
a(n) = A025035(n)*n! - Geoffrey Critzer, Jun 07 2014
a(n) = A089759(3,n). - R. J. Mathar, Nov 01 2015
From Amiram Eldar, Jan 26 2022: (Start)
Sum_{n>=0} 1/a(n) = (exp(6^(1/3)) + 2*exp(-6^(1/3)/2)*cos(3^(5/6)/2^(2/3)))/3.
Sum_{n>=0} (-1)^n/a(n) = (exp(-6^(1/3)) + 2*exp(6^(1/3)/2)*cos(3^(5/6)/2^(2/3)))/3. (End)

A034841 a(n) = (n^2)! / (n!)^n.

Original entry on oeis.org

1, 1, 6, 1680, 63063000, 623360743125120, 2670177736637149247308800, 7363615666157189603982585462030336000, 18165723931630806756964027928179555634194028454000000, 53130688706387569792052442448845648519471103327391407016237760000000000

Offset: 0

Erich Friedman

nonn

The number of arrangements of 1,2,...,n^2 in an n X n matrix such that each row is increasing. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 12 2001
a(n) == 0 (mod n!). In fact (n^2)! == 0 (mod (n!)^n) by elementary combinatorics, a better result is (n^2)! == 0 (mod (n!)^(n+1)). - Amarnath Murthy, Jul 13 2005
a(n) is also the number of lattice paths from {n}^n to {0}^n using steps that decrement one component by 1. a(2) = 6: [(2,2), (1,2), (0,2), (0,1), (0,0)], [(2,2), (1,2), (1,1), (0,1), (0,0)], [(2,2), (1,2), (1,1), (1,0), (0,0)], [(2,2), (2,1), (1,1), (0,1), (0,0)], [(2,2), (2,1), (1,1), (1,0), (0,0)], [(2,2), (2,1), (2,0), (1,0), (0,0)]. - Alois P. Heinz, May 06 2013
Given n^2 distinguishable balls and n distinguishable urns, a(n) = the number of ways to place n balls in the i-th urn for all 1 <= i <= n, where n = n_1 + n_2 + ... + n_n. - Ross La Haye, Dec 28 2013

Alois P. Heinz and Tilman Piesk, Table of n, a(n) for n = 0..26 (first 20 terms from Alois P. Heinz)
Noah Lordi, Maedee Trank-Greene, Akira Kyle, and Joshua Combes, Quantum permutation puzzles with indistinguishable particles, arXiv:2410.22287 [quant-ph], 2024. See p. 8.

Cf. A000142, A039622, A229050, A229050, A340590.
Cf. A000984, A006480, A008977, A008978, A022915.
Main diagonal of A089759, A187783.

[Factorial(n^2) / Factorial(n)^n: n in [0..10]]; // Vincenzo Librandi, Oct 29 2014

a:= n-> (n^2)! / (n!)^n:
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 24 2012

Prepend[Table[nn = n^2;nn! Coefficient[Series[(x^n/n!)^n, {x, 0, nn}], x^nn], {n, 1, 15}], 1] (* Geoffrey Critzer, Mar 08 2015 *)

a(n) = (n^2)! / (n!)^n; \\ Michel Marcus, Oct 28 2014

Using a higher order version of Stirling's formula (the "standard" formula appears in A000142) we have the asymptotic expression: a(n) ~ sqrt(2*Pi) * e^(-1/12) * n^(n^2 - n/2 + 1) / (2*Pi)^(n/2). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 13 2001

a(n) = Product_{k=1..n} binomial(k*n, n). - Vaclav Kotesovec, Mar 10 2019

a(0)=1 prepended by Tilman Piesk, Oct 28 2014

A187783 De Bruijn's triangle, T(m,n) = (m*n)!/(n!^m) read by downward antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 6, 1, 1, 1, 20, 90, 24, 1, 1, 1, 70, 1680, 2520, 120, 1, 1, 1, 252, 34650, 369600, 113400, 720, 1, 1, 1, 924, 756756, 63063000, 168168000, 7484400, 5040, 1

Offset: 0

Robert G. Wilson v, Jan 05 2013

From Tilman Piesk, Oct 28 2014: (Start)
Number of permutations of a multiset that contains m different elements n times. These multisets have the signatures A249543(m,n-1) for m>=1 and n>=2.
In an m-dimensional Pascal tensor (the generalization of a symmetric Pascal matrix) P(x1,...,xn) = (x1+...+xn)!/(x1!*...*xn!), so the main diagonal of an m-dimensional Pascal tensor is D(n) = (m*n)!/(n!^m). These diagonals are the rows of this array (with m>0), which begins like this:
  m\n:0   1       2            3                4                    5
  0:  1   1       1            1                1                    1 ... A000012;
  1:  1   1       1            1                1                    1 ... A000012;
  2:  1   2       6           20               70                  252 ... A000984;
  3:  1   6      90         1680            34650               756756 ... A006480;
  4:  1  24    2520       369600         63063000          11732745024 ... A008977;
  5:  1 120  113400    168168000     305540235000      623360743125120 ... A008978;
  6:  1 720 7484400 137225088000 3246670537110000 88832646059788350720 ... A008979;
with columns: A000142 (n=1), A000680 (n=2), A014606 (n=3), A014608 (n=4), A014609 (n=5).
A089759 is the transpose of this matrix. A034841 is its diagonal. A141906 is its lower triangle. A120666 is the upper triangle of this matrix with indices starting from 1. A248827 are the diagonal sums (or the row sums of the triangle).
(End)

			T(3,5) = (3*5)!/(5!^3) = 756756 = A014609(3) = A006480(5) is the number of permutations of a multiset that contains 3 different elements 5 times, e.g., {1,1,1,1,1,2,2,2,2,2,3,3,3,3,3}.

Tilman Piesk, First 54 rows of the triangle, flattened
T. Chappell, A. Lascoux, S. Ole Warnaar, and W. Zudilin, Logarithmic and complex constant term identities, arXiv:1112.3130 [math.CO], 2012.
Tilman Piesk, Array for indices 0..16
Tilman Piesk, PHP code used to create the b-file
Tilman Piesk, Illustration of the multisets for m,n=0..4
Wikipedia, Permutations of multisets, Pascal matrix and simplex

Cf. A089759 (transposed), A141906 (subtriangle), A120666 (subtriangle transposed), A060538 (1st row/column removed).
Rows: A000012, A000984, A006480, A006480, A008978, A008979, A071549, A071550, A071551, A071552.
Columns: A000012, A000142, A000680, A014606, A014608, A014609, A248814, A172603, A172609, A172613.
Main diagonal gives: A034841.
Row sums of the triangle: A248827.

[Factorial(k*(n-k))/(Factorial(n-k))^k: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 26 2022

T[n_, k_]:= (k*n)!/(n!)^k; Table[T[n, k-n], {k, 9}, {n, 0, k-1}]//Flatten

def A187783(n,k): return gamma(k*(n-k)+1)/(factorial(n-k))^k
flatten([[A187783(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Dec 26 2022

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

A060540(m,n) = T(m,n)/m! . - R. J. Mathar, Jun 21 2023

Row m=0 prepended by Tilman Piesk, Oct 28 2014

A269129 Number A(n,k) of sequences with k copies each of 1,2,...,n avoiding the pattern 12...n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 5, 1, 0, 0, 1, 43, 23, 1, 0, 0, 1, 374, 1879, 119, 1, 0, 0, 1, 3199, 173891, 102011, 719, 1, 0, 0, 1, 26945, 16140983, 117392909, 7235651, 5039, 1, 0, 0, 1, 224296, 1474050783, 142951955371, 117108036719, 674641325, 40319, 1

Offset: 0

Alois P. Heinz, Feb 19 2016

			Square array A(n,k) begins:
  0,   0,      0,         0,            0,               0, ...
  1,   0,      0,         0,            0,               0, ...
  1,   1,      1,         1,            1,               1, ...
  1,   5,     43,       374,         3199,           26945, ...
  1,  23,   1879,    173891,     16140983,      1474050783, ...
  1, 119, 102011, 117392909, 142951955371, 173996758190594, ...

Alois P. Heinz, Antidiagonals n = 0..50, flattened
J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Columns k=0-10 give: A057427, A033312, A267532, A269113, A269114, A269115, A269116, A269117, A269118, A269119, A269120.
Rows n=0-10 give: A000004, A000007, A000012, A269121, A269122, A269123, A269124, A269125, A269126, A269127, A269128.
Main diagonal gives: A268751.
Cf. A047909, A089759, A187783, A331562.

g:= proc(l) option remember; (n-> f(l[1..nops(l)-1])*
      binomial(n-1, l[-1]-1)+add(f(sort(subsop(j=l[j]
      -1, l))), j=1..nops(l)-1))(add(i, i=l))
    end:
f:= l->(n->`if`(n=0, 1, `if`(l[1]=0, 0, `if`(n=1 or l[-1]=1, 1,
    `if`(n=2, binomial(l[1]+l[2], l[1])-1, g(l))))))(nops(l)):
A:= (n, k)-> (k*n)!/k!^n - f([k$n]):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
b:= proc(k, p, j, l, t) option remember;
      `if`(k=0, (-1)^t/l!, `if`(p<0, 0, add(b(k-i, p-1,
       j+1, l+i*j, irem(t+i*j, 2))/(i!*p!^i), i=0..k)))
    end:
A:= (n, k)-> (n*k)!*(1/k!^n-b(n, k-1, 1, 0, irem(n, 2))*n!):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 03 2016

b[k_, p_, j_, l_, t_] := b[k, p, j, l, t] = If[k == 0, (-1)^t/l!, If[p < 0, 0, Sum[b[k-i, p-1, j+1, l + i j, Mod[t + i j, 2]]/(i! p!^i), {i, 0, k}]] ];
A[n_, k_] := (n k)! (1/k!^n - b[n, k-1, 1, 0, Mod[n, 2]] n!); Table[ Table[ A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Apr 07 2016, after Alois P. Heinz *)

A(n,k) = A089759(k,n) - A047909(k,n) = A187783(n,k) - A047909(k,n).

A225094 Number A(n,k) of lattice paths without interior points from {n}^k to {0}^k using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 6, 2, 0, 1, 1, 24, 54, 2, 0, 1, 1, 120, 1944, 384, 2, 0, 1, 1, 720, 99000, 132000, 2550, 2, 0, 1, 1, 5040, 6966000, 79716000, 8059800, 16506, 2, 0, 1, 1, 40320, 655678800, 78928416000, 57010275000, 471369024, 105840, 2, 0, 1

Offset: 0

Alois P. Heinz, Apr 27 2013

An interior point p = (p_1, ..., p_k) has k>0 components with 0

			A(n,0) = 1: [()].
A(0,k) = 1: [{0}^k].
A(1,1) = 1: [(1), (0)].
A(2,1) = 0, there is no path from (2) to (0) without interior points.
A(1,2) = 2: [(1,1), (0,1), (0,0)], [(1,1), (1,0), (0,0)].
A(1,3) = 6: [(1,1,1), (0,1,1), (0,0,1), (0,0,0)], [(1,1,1), (0,1,1), (0,1,0), (0,0,0)], [(1,1,1), (1,0,1), (0,0,1), (0,0,0)], [(1,1,1), (1,0,1), (1,0,0), (0,0,0)], [(1,1,1), (1,1,0), (0,1,0), (0,0,0)], [(1,1,1), (1,1,0), (1,0,0), (0,0,0)].
Square array A(n,k) begins:
  1, 1, 1,     1,         1,              1, ...
  1, 1, 2,     6,        24,            120, ...
  1, 0, 2,    54,      1944,          99000, ...
  1, 0, 2,   384,    132000,       79716000, ...
  1, 0, 2,  2550,   8059800,    57010275000, ...
  1, 0, 2, 16506, 471369024, 38606650125120, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0, 2-4 give: A000012, A040000, A060774, A225220.
Rows n=0-4 give: A000012, A000142, A071798(k) (for k>0), A225096, A225221.
Main diagonal gives: A225111.
Cf. A089759 (unrestricted paths), A210472, A262809, A263159.

b:= proc(n, l) option remember; local m; m:= nops(l);
      `if`(m=0 or l[m]=0, 1, `if`(l[1]>0 and l[m] b(n, [n$k]):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, l_] := b[n, l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[l[[1]] > 0 && l[[m]] < n, 0, Sum[If[l[[i]] == 0, 0, b[n, Sort[ReplacePart[l, i -> l[[i]] - 1]]]], {i, 1, m}]]] ]; a[n_, k_] := b[n, Array[n&, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

A210472 Number A(n,k) of paths starting at {n}^k to a border position where one component equals 0 using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 33, 20, 1, 0, 1, 5, 196, 543, 70, 1, 0, 1, 6, 1305, 22096, 10497, 252, 1, 0, 1, 7, 9786, 1304045, 3323092, 220503, 924, 1, 0, 1, 8, 82201, 106478916, 1971644785, 574346824, 4870401, 3432, 1, 0

Offset: 0

Alois P. Heinz, Jan 22 2013

			A(0,3) = 1: [(0,0,0)].
A(1,1) = 1: [(1), (0)].
A(1,2) = 2: [(1,1), (0,1)], [(1,1), (1,0)].
A(1,3) = 3: [(1,1,1), (0,1,1)], [(1,1,1), (1,0,1)], [(1,1,1), (1,1,0)].
A(2,1) = 1: [(2), (1), (0)].
A(2,2) = 6: [(2,2), (1,2), (0,2)], [(2,2), (1,2), (1,1), (0,1)], [(2,2), (1,2), (1,1), (1,0)], [(2,2), (2,1), (1,1), (0,1)], [(2,2), (2,1), (1,1), (1,0)], [(2,2), (2,1), (2,0)].
Square array A(n,k) begins:
  0, 1,   1,      1,         1,             1, ...
  0, 1,   2,      3,         4,             5, ...
  0, 1,   6,     33,       196,          1305, ...
  0, 1,  20,    543,     22096,       1304045, ...
  0, 1,  70,  10497,   3323092,    1971644785, ...
  0, 1, 252, 220503, 574346824, 3617739047205, ...

Alois P. Heinz, Antidiagonals n = 0..24, flattened

Columns k=0-4 give: A000004, A000012, A000984, A209245, A209288.
Rows n=0-3 give: A057427, A001477, A093964, A210486.
Main diagonal gives A276490.
Cf. A089759 (unrestricted paths), A225094, A262809, A263159.

b:= proc() option remember; `if`(nargs=0, 0, `if`(args[1]=0, 1,
      add(b(sort(subsop(i=args[i]-1, [args]))[]), i=1..nargs)))
    end:
A:= (n, k)-> b(n$k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[] = 0; b[args__] := b[args] = If[First[{args}] == 0, 1, Sum[b @@ Sort[ReplacePart[{args}, i -> {args}[[i]] - 1]], {i, 1, Length[{args}]}]]; a[n_, k_] := b @@ Array[n&, k]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

A071549 a(n) = (7n)!/n!^7.

Original entry on oeis.org

1, 5040, 681080400, 182509367040000, 66475579247327250000, 28837919555681211870935040, 14007180988362844601443040716800, 7363615666157189603982585462030336000, 4104167472585675600759440022842715359250000, 2392741010223442438553822446842770682716580000000

Offset: 0

Benoit Cloitre, May 30 2002

nonn

Number of closed paths of length 7n whose steps are 7th roots of unity. - Andrew Howroyd, Nov 01 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Row n=7 of A187783, column k=7 of A089759.

Sequences (k*n)!/n!^k: A000984 (k = 2), A006480 (k =3), A008977 (k = 4), A008978 (k = 5), A008979 (k = 6), A071550 (k = 8), A071551 (k = 9), A071552 (k = 10).

[Factorial(7*n)/Factorial(n)^7: n in [0..20]]; // Vincenzo Librandi, Aug 13 2014

Table[(7 n)!/(n)!^7, {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2014 *)

a(n) = (7*n)!/(n!^7); \\ Andrew Howroyd, Nov 01 2018

From Peter Bala, Feb 14 2020: (Start)
a(n) = C(7*n,n)*C(6*n,n)*C(5*n,n)*C(4*n,n)*C(3*n,n)*C(2*n,n).
a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers m and k - apply  Mestrovic, Equation 39, p. 12.
a(n) = [x^n](F(x)^(5040*n)), where F(x) = 1 + x + 62528*x^2 + 11087269661*x^3 + 3021437267047869*x^4 + 1045823730475703710735*x^5 + ...
appears to have integer coefficients. For similar results see A008979.
a(n) = [(x*y*z*u*v*w)^n] (1 + x + y + z + u + v + w)^(7*n). (End)

a(8)-a(9) added by Andrew Howroyd, Nov 01 2018

A347811 Number A(n,k) of k-dimensional lattice walks from {n}^k to {0}^k using steps that decrease the Euclidean distance to the origin and that change each coordinate by at most 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 19, 25, 1, 1, 1, 323, 211075, 241, 1, 1, 1, 38716, 1322634996717, 2062017739, 2545, 1, 1, 1, 32253681, 16042961630858858915656, 29261778984922904560001, 32191353922714, 28203, 1, 1

Offset: 0

Alois P. Heinz, Sep 14 2021

Lattice points may have negative coordinates, and different walks may differ in length. All walks are self-avoiding.

			Square array A(n,k) begins:
  1, 1,     1,              1,                       1,     1, ...
  1, 1,     3,             19,                     323, 38716, ...
  1, 1,    25,         211075,           1322634996717, ...
  1, 1,   241,     2062017739, 29261778984922904560001, ...
  1, 1,  2545, 32191353922714, ...
  1, 1, 28203, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..11, flattened
Alois P. Heinz, Animation of A(2,2) = 25 walks
Wikipedia, Counting lattice paths
Wikipedia, Self-avoiding walk

Columns k=0+1, 2-3 give: A000012, A346539, A347813.
Rows n=0-2 give: A000012, A346840, A347812.
Main diagonal gives A347810.
Cf. A089759, A187783, A335570.

s:= proc(n) option remember;
     `if`(n=0, [[]], map(x-> seq([x[], i], i=-1..1), s(n-1)))
    end:
b:= proc(l) option remember; (n-> `if`(l=[0$n], 1, add((h-> `if`(
      add(i^2, i=h) b([n$k]):
seq(seq(A(n, d-n), n=0..d), d=0..8);

s[n_] := s[n] = If[n == 0, {{}}, Sequence @@ Table[Append[#, i], {i, -1, 1}]& /@ s[n-1]];
b[l_List] := b[l] = With[{n = Length[l]}, If[l == Table[0, {n}], 1, Sum[With[{h = l+x}, If[h.h < l.l, b[Sort[h]], 0]], {x, s[n]}]]];
A[n_, k_] := b[Table[n, {k}]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Nov 03 2021, after Alois P. Heinz *)

Showing 1-10 of 17 results. Next

cp's OEIS Frontend

A248827 Row sums of A187783 and A089759.

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A000680 a(n) = (2n)!/2^n.

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A014606 a(n) = (3n)!/(6^n).

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A034841 a(n) = (n^2)! / (n!)^n.

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A187783 De Bruijn's triangle, T(m,n) = (m*n)!/(n!^m) read by downward antidiagonals.

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A269129 Number A(n,k) of sequences with k copies each of 1,2,...,n avoiding the pattern 12...n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A225094 Number A(n,k) of lattice paths without interior points from {n}^k to {0}^k using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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