A001286 -id:A001286 - cp's OEIS frontend

A281881 Triangle read by rows: T(n,k) (n>=1, 2<=k<=n+1) is the number of k-sequences of balls colored with at most n colors such that exactly one ball is of a color seen previously in the sequence.

1, 2, 6, 3, 18, 36, 4, 36, 144, 240, 5, 60, 360, 1200, 1800, 6, 90, 720, 3600, 10800, 15120, 7, 126, 1260, 8400, 37800, 105840, 141120, 8, 168, 2016, 16800, 100800, 423360, 1128960, 1451520

Offset: 1

Jeremy Dover, Feb 01 2017

Number of k-sequences of balls colored with at most n colors such that exactly two balls are the same color as some other ball in the sequence (necessarily each other). - Jeremy Dover, Sep 26 2017

			n=1 => AA -> T(1,2) = 1.
n=2 => AA, BB -> T(2,2) = 2; AAB, ABA, BAA, BBA, BAB, ABB -> T(2,3) = 6.
Triangle starts:
   1
   2,   6
   3,  18,   36
   4,  36,  144,   240
   5,  60,  360,  1200,   1800
   6,  90,  720,  3600,  10800,   15120
   7, 126, 1260,  8400,  37800,  105840,   141120
   8, 168, 2016, 16800, 100800,  423360,  1128960,  1451520
   9, 216, 3024, 30240, 226800, 1270080,  5080320, 13063680,  16329600
  10, 270, 4320, 50400, 453600, 3175200, 16934400, 65318400, 163296000, 199584000

Jeremy Dover, Table of n, a(n) for n = 1..999
Jeremy Dover, Answer to Cumulative Distribution Function of Collision Counts

Columns of table:
  T(n,2) = A000027(n)
  T(n,3) = A028896(n)
Other sequences in table:
  T(n,n+1) = A001286(n)
  T(n,n) = A001804(n), n>=2

Table[Binomial[k, 2] n!/(n + 1 - k)!, {n, 8}, {k, 2, n + 1}] // Flatten (* Michael De Vlieger, Feb 02 2017 *)

T(n,k) = binomial(k,2)*n!/(n+1-k)!.

T(n,k) = n*T(n-1,k-1) + (k-1)*n!/(n+1-k)!.

A300559 a(n) = n*(n+1)!/2 + 1.

Original entry on oeis.org

1, 2, 7, 37, 241, 1801, 15121, 141121, 1451521, 16329601, 199584001, 2634508801, 37362124801, 566658892801, 9153720576001, 156920924160001, 2845499424768001, 54420176498688001, 1094805903679488001, 23112569077678080001, 510909421717094400001, 11802007641664880640001

Offset: 0

M. F. Hasler, Apr 10 2018

See A301373 and A302859 for the primes: it is remarkable that all of a(1..10) are primes, and only a(11) is the first composite term.

Maheswara Rao Valluri, Primes of the form p = 1 + n! Sum n, for some n ∈ N*, arXiv:1803.11461 [math.GM], 2018.
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.

Inspired by A302859.

Cf. A301373.

[n*Factorial(n+1)/2+1: n in [0..25]]; // Vincenzo Librandi, Apr 12 2018

Array[# (# + 1)!/2 + 1 &, 22, 0] (* Michael De Vlieger, Apr 10 2018 *)

apply( A300559=n->(n+1)!\2*n+1, [0..25])

a(n) = A180119(n) + 1 = A001286(n+1) + 1.
D-finite with recurrence n*a(n+1) = (n+1)*(n+2)*(a(n)-1) + n. - Chai Wah Wu, Apr 11 2018
E.g.f.: exp(x)-1/(x-1)^3*x. - Simon Plouffe, Jun 21 2018

A301373 Numbers k such that (k+1)!*k/2 + 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 19, 24, 251, 374, 953, 1104, 1507, 3390, 4443, 5762

Offset: 1

Daniel Suteu, Apr 03 2018

The associated primes are A300559(a(n)) = A180119(a(n))+1 = A001286(a(n)+1)+1. - M. F. Hasler, Apr 10 2018
Looking for primes of the form p(n) = 1 + n! f(n) with a simple polynomial function f, it appears that the choice f(n) = n(n+1)/2 = A000217 is one of the most successful choices for getting a maximum of primes for n = 1..20. - M. F. Hasler, Apr 14 2018
The PFGW program has been used to certify all the terms up to a(23), using a deterministic test which exploits the factorization of a(n) - 1. - Giovanni Resta, Jun 24 2018

Maheswara Rao Valluri, Primes of the form p = 1 + n! Sum n, for some n ∈ N*, arXiv:1803.11461 [math.GM], 2018.

Cf. A090703, A300559, A180119, A001286.

See A302859 for the actual primes.

Do[ If[ PrimeQ[n(n +1)!/2 +1], Print@ n], {n, 4000}] (* Robert G. Wilson v, Apr 05 2018 *)

isok(k) = ispseudoprime((k+1)! * k / 2 + 1);

a(21) from Robert G. Wilson v, Apr 05 2018
a(22) from Vaclav Kotesovec, Apr 06 2018
a(23) from Giovanni Resta, Jun 24 2018

A060570 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=2 and D varies.

Original entry on oeis.org

0, 1, 8, 100, 2144, 80360

Offset: 2

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

nonn

			For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 1.

A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices, arXiv:math/0008022 [math.CO], 2000.

Cf. A001286 (case where d=1), A006245 (number of 2-tilings). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

A060608 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=3 and D varies.

Original entry on oeis.org

0, 1, 10, 264, 22624

Offset: 3

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

nonn

			For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 1.

A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices, arXiv:math/0008022 [math.CO], 2000.

Cf. A001286 (case where d=1). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

A060612 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=4 and D varies.

Original entry on oeis.org

0, 1, 12, 672

Offset: 5

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

nonn

			For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 1.

A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices

Cf. A001286 (case where d=1). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

A152883 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which k is an excedance (n >= 2, 1 <= k <= n-1). An excedance of a permutation p is a value j such that p(j) > j.

Original entry on oeis.org

1, 4, 2, 18, 12, 6, 96, 72, 48, 24, 600, 480, 360, 240, 120, 4320, 3600, 2880, 2160, 1440, 720, 35280, 30240, 25200, 20160, 15120, 10080, 5040, 322560, 282240, 241920, 201600, 161280, 120960, 80640, 40320, 3265920, 2903040, 2540160, 2177280, 1814400, 1451520, 1088640, 725760, 362880

Offset: 2

Emeric Deutsch, Jan 13 2009

Sum of entries in row n = n!*(n-1)/2 = A001286(n) (the Lah numbers).

T(n,n-1) = (n-1)! (A000142).

			T(4,3) = 6 because the permutations of {1,2,3,4} in which 3 is an excedance are 1243, 1342, 3142, 2143, 2341 and 3241.
Triangle starts:
    1;
    4,   2;
   18,  12,   6;
   96,  72,  48,  24;
  600, 480, 360, 240, 120;

Cf. A000142, A001286.

T := proc (n, k) options operator, arrow: factorial(n-1)*(n-k) end proc: for n from 2 to 10 do seq(T(n, k), k = 1 .. n-1) end do;

T(n,k) = (n-1)!*(n-k) (n >= 2, 1 <= k <= n-1). [Proof: n-k choices for p(k) and (n-1)! choices for the remaining entries of p.]

A193561 Augmentation of the triangle A004736. See Comments.

Original entry on oeis.org

1, 2, 1, 6, 6, 3, 24, 36, 30, 15, 120, 240, 270, 210, 105, 720, 1800, 2520, 2520, 1890, 945, 5040, 15120, 25200, 30240, 28350, 20790, 10395, 40320, 141120, 272160, 378000, 415800, 374220, 270270, 135135, 362880, 1451520, 3175200, 4989600

Offset: 0

Clark Kimberling, Jul 30 2011

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding A193561, if the triangle is written as (w(n,k)), then
w(n,n)=A001147(n), "double factorial numbers";
w(n,n-1)=A097801(n), (2n)!/(n!*2^(n-1))
col 1:  A000142, n!
col 2: A001286, Lah numbers, (n-1)*n!/2

			First 5 rows of A193560:
1
2.....1
6.....6....3
24....36...30...15
120...240..270..210..105

Cf. A193091.

p[n_, k_] := n + 1 - k
Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* A004736 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]]  (* A193561 *)
Flatten[Table[v[n], {n, 0, 8}]]

A213168 a(n) = n!/2 - (n-1)! - n + 2.

Original entry on oeis.org

0, 0, 4, 33, 236, 1795, 15114, 141113, 1451512, 16329591, 199583990, 2634508789, 37362124788, 566658892787, 9153720575986, 156920924159985, 2845499424767984, 54420176498687983, 1094805903679487982, 23112569077678079981, 510909421717094399980

Offset: 2

Olivier Gérard, Nov 02 2012

Row sums of A142706 for k=1..n-1.

Vincenzo Librandi, Table of n, a(n) for n = 2..200

Cf. A001286.
Cf. A142706, A051683, A152883.
Cf. A200748 (considered as a triangular array).

[Factorial(n)/2-Factorial(n-1)-n+2: n in [2..25]]; // Vincenzo Librandi, Sep 09 2016

f:=gfun:-rectoproc({2*(n-3)*a(n) - (2*n^2-6*n+4)*a(n-1)- 2*(n-3)*(n-2)^2, a(2)=0,a(3)=0},a(n),remember): map(f, [$2..22]); # Georg Fischer, Aug 25 2021

Table[n!/2 - (n - 1)! - n + 2, {n, 2, 20}]

A213168(n):=n!/2-(n-1)!-n+2$
makelist(A213168(n),n,2,30); /* Martin Ettl, Nov 03 2012 */

a(n) = A001286(n-1) - n + 2. - Anton Zakharov, Sep 08 2016
D-finite with recurrence: 2*(n-3)*a(n) - (2*n^2-6*n+4)*a(n-1)- 2*(n-3)*(n-2)^2 = 0. - Georg Fischer, Aug 25 2021
E.g.f.: 1/(2-2*x)+log(1-x)+(2-x)*exp(x). - Alois P. Heinz, Aug 25 2021

A232433 E.g.f. satisfies: A(x,q) = exp( Integral A(x,q)A(qx,q) dx ).

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 2, 1, 24, 36, 22, 14, 6, 2, 1, 120, 240, 210, 160, 104, 56, 32, 14, 6, 2, 1, 720, 1800, 2040, 1830, 1448, 992, 674, 408, 232, 128, 68, 32, 14, 6, 2, 1, 5040, 15120, 21000, 21840, 19824, 15834, 12144, 8758, 5904, 3860, 2442, 1482, 870, 492, 260, 142, 68, 32, 14, 6, 2, 1

Offset: 0

Paul D. Hanna, Nov 23 2013

			E.g.f.: A(x,q) = 1 + (1)*x + (2 + q)*x^2/2! + (6 + 6*q + 2*q^2 + q^3)*x^3/3!
+ (24 + 36*q + 22*q^2 + 14*q^3 + 6*q^4 + 2*q^5 + q^6)*x^4/4!
+ (120 + 240*q + 210*q^2 + 160*q^3 + 104*q^4 + 56*q^5 + 32*q^6 + 14*q^7 + 6*q^8 + 2*q^9 + q^10)*x^5/5! +...
The triangle of coefficients T(n,k) of x^n*q^k, for n>=0, k=0..n*(n-1)/2, in e.g.f. A(x,q) begins:
[1];
[1];
[2, 1];
[6, 6, 2, 1];
[24, 36, 22, 14, 6, 2, 1];
[120, 240, 210, 160, 104, 56, 32, 14, 6, 2, 1];
[720, 1800, 2040, 1830, 1448, 992, 674, 408, 232, 128, 68, 32, 14, 6, 2, 1];
[5040, 15120, 21000, 21840, 19824, 15834, 12144, 8758, 5904, 3860, 2442, 1482, 870, 492, 260, 142, 68, 32, 14, 6, 2, 1];
[40320, 141120, 231840, 275520, 280056, 251496, 212112, 170424, 129716, 95248, 67632, 46616, 31280, 20576, 13142, 8232, 5004, 2954, 1706, 966, 524, 276, 142, 68, 32, 14, 6, 2, 1]; ...
The limit of the reversed rows (A232434) begins:
[1, 2, 6, 14, 32, 68, 142, 276, 542, 1022, 1876, 3394, 6066, 10628, ...].

Paul D. Hanna, Table of n, a(n) for n = 0..9919; rows 0..39 in flattened triangle.
Miguel A. Mendez, Combinatorial differential operators in: Faà di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees, arXiv:1610.03602 [math.CO], 2016.

Cf. A232434, A126470, A001286, A075183.

nmax = 8; A[, ] = 0; Do[A[x_, q_] = Exp[Integrate[A[x, q] A[q x, q], x]] + O[x]^n // Normal // Simplify, {n, nmax}];
CoefficientList[#, q]& /@ (CoefficientList[A[x, q], x] Range[0, nmax-1]!) // Flatten (* Jean-François Alcover, Oct 27 2018 *)

{T(n,k)=local(A=1+x);for(i=1,n,A=exp(intformal(A*subst(A,x,x*y +x*O(x^n)),x)));n!*polcoeff(polcoeff(A,n,x),k,y)}
for(n=0,12,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))

E.g.f. satisfies: d/dx A(x,q) = A(x,q)^2 * A(q*x,q).
Row sums equal the odd double factorials.
Limit of reversed rows yield A232434.

cp's OEIS Frontend

A281881 Triangle read by rows: T(n,k) (n>=1, 2<=k<=n+1) is the number of k-sequences of balls colored with at most n colors such that exactly one ball is of a color seen previously in the sequence.

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A300559 a(n) = n*(n+1)!/2 + 1.

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Magma

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A301373 Numbers k such that (k+1)!*k/2 + 1 is prime.

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A060570 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=2 and D varies.

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A060608 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=3 and D varies.

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A060612 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=4 and D varies.

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A152883 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which k is an excedance (n >= 2, 1 <= k <= n-1). An excedance of a permutation p is a value j such that p(j) > j.

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Maple

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A193561 Augmentation of the triangle A004736. See Comments.

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Mathematica

A213168 a(n) = n!/2 - (n-1)! - n + 2.

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A232433 E.g.f. satisfies: A(x,q) = exp( Integral A(x,q)A(qx,q) dx ).