Jeremy Dover - cp's OEIS frontend

A292999 Triangle read by rows: T(n,k) (n >= 1, 4 <= k <= n+3) is the number of k-sequences of balls colored with at most n colors such that exactly four balls are the same color as some other ball in the sequence.

1, 8, 10, 21, 120, 90, 40, 420, 1440, 840, 65, 1000, 6300, 16800, 8400, 96, 1950, 18000, 88200, 201600, 90720, 133, 3360, 40950, 294000, 1234800, 2540160, 1058400, 176, 5320, 80640, 764400, 4704000, 17781120, 33868800, 13305600, 225, 7920, 143640, 1693440, 13759200, 76204800, 266716800, 479001600, 179625600

Offset: 1

Jeremy Dover, Sep 27 2017

			For n=1: AAAA -> T(1,4)=1.
For n=2: AAAA,BBBB,AABB,ABAB,ABBA,BAAB,BABA,BBAA -> T(2,4)=8; AAAAB,AAABA,AABAA,ABAAA,BAAAA,BBBBA,BBBAB,BBABB,BABBB,ABBBB -> T(2,5)=10.
Triangle starts:
    1;
    8,   10;
   21,  120,     90;
   40,  420,   1440,     840;
   65, 1000,   6300,   16800,     8400;
   96, 1950,  18000,   88200,   201600,    90720;
  133, 3360,  40950,  294000,  1234800,  2540160,   1058400;
  176, 5320,  80640,  764400,  4704000, 17781120,  33868800, 13305600;
  225, 7920, 143640, 1693440, 13759200, 76204800, 266716800, ... .

Columns of the table: T(n,4) = A000567(n), T(n,5) = 10*A007586(n-1), T(n,6) = 90*A220212(n-2).

Diagonals of the table: T(n,n+3) = A061206(n), T(n+1,n+3) = 8*A005461(n), T(n-1,n) = 21*A001755(n), T(n,n) = 40*A001811(n), T(n,n-1) = 65*A001777(n), T(n+6,n+4) = A062194(n).

Table[Binomial[k, 4] n! (1/(n + 3 - k)! + 3/(n + 2 - k)!), {n, 9}, {k, 4, n + 3}] // Flatten (* Michael De Vlieger, Sep 30 2017 *)

a(n) = binomial(k,4)*n!*(1/(n+3-k)! + 3/(n+2-k)!) (with the convention that 3/(-1)! = 0 when k=n+3).

A292998 Number of sequences of balls colored with at most n colors such that exactly three balls are the same color as some other ball in the sequence.

Original entry on oeis.org

1, 10, 87, 772, 7285, 74046, 812875, 9626632, 122643657, 1675253170, 24449818591, 379984902540, 6268557335677, 109443030279142, 2016658652491155, 39119860206021136, 797013832285599505, 17017679492994949722, 380045072079456330727

Offset: 1

Jeremy Dover, Sep 27 2017

nonn

Note that any such sequence has at least 3 balls and at most n+2, and that three matching balls must all be the same color.

			For n=2 colors a, b, the a(n)=10 sequences of balls are: aaa, bbb, abbb, babb, bbab, bbba, baaa, abaa, aaba, aaab.

Jeremy M. Dover, Repetition in Colored Sequences of Balls, arXiv:1710.06049 [math.CO], 2017.

Row sums of triangle A292930.

Cf. A281912, A292999.

Table[n!*Sum[Binomial[k, 3]/(n + 2 - k)!, {k, 3, n + 2}], {n, 19}] (* Michael De Vlieger, Sep 28 2017 *)

a(n) = n! * sum(k=3, n+2, binomial(k,3)/(n+2-k)!); \\ Michel Marcus, Sep 29 2017

a(n) = n! * Sum_{k=3..n+2} binomial(k,3)/(n+2-k)!.

A292930 Triangle read by rows: T(n,k) (n>=1, 3<=k<=n+2) is the number of k-sequences of balls colored with at most n colors such that exactly three balls are the same color as some other ball in the sequence.

Original entry on oeis.org

1, 2, 8, 3, 24, 60, 4, 48, 240, 480, 5, 80, 600, 2400, 4200, 6, 120, 1200, 7200, 25200, 40320, 7, 168, 2100, 16800, 88200, 282240, 423360, 8, 224, 3360, 33600, 235200, 1128960, 3386880, 4838400, 9, 288, 5040, 60480, 529200, 3386880, 15240960, 43545600, 59875200, 10, 360, 7200, 100800, 1058400, 8467200, 50803200, 217728000, 598752000, 798336000

Offset: 1

Jeremy Dover, Sep 26 2017

Note that the three matching balls are necessarily the same color.

			n=1 => AAA -> T(1,3)=1;
n=2 => AAA,BBB -> T(2,3)=2;
       AAAB,AABA,ABAA,BAAA,BBBA,BBAB,BABB,ABBB -> T(2,4)=8.
Triangle begins:
  1;
  2, 8;
  3, 24, 60;
  4, 48, 240, 480;
  5, 80, 600, 2400, 4200;
  ...

Columns of table: T(n,3) = A000027(n), T(n,4) = A033996(n).

Other sequences in table: T(n,n+2) = A005990(n+1).

T(n, k) = binomial(k,3)*n!/(n+2-k)!;
tabl(nn) = for (n=1, nn, for (k=3, n+2, print1(T(n,k), ", ")); print()); \\ Michel Marcus, Sep 29 2017

T(n, k) = binomial(k,3)*n!/(n+2-k)!.

A292880 Number of sequences of balls colored with at most n colors such that exactly three balls are of a color seen earlier in the sequence.

Original entry on oeis.org

1, 32, 633, 10744, 173705, 2798376, 45930577, 777101648, 13638044529, 249079033360, 4741200949001, 94104123729672, 1947270419971513, 41985753920469464, 942531024150018465, 22009425078894009376, 534085741053864862817, 13454221423402868473728, 351483652960252663137049, 9512821482149972773978520

Offset: 1

Jeremy Dover, Sep 25 2017

nonn

Note that any such sequence has at least 4 balls and at most n+3.

Row sums of triangle A292879.

a(n) = n! * Sum_{k=4..n+3} [binomial(k,4)+10*binomial(k,5)+15*binomial(k,6)]/(n+3-k)!

A292879 Triangle read by rows: T(n,k) (n>=1, 4<=k<=n+3) is the number of k-sequences of balls colored with at most n colors such that exactly three balls are of a color seen previously in the sequence.

Original entry on oeis.org

1, 2, 30, 3, 90, 540, 4, 180, 2160, 8400, 5, 300, 5400, 42000, 126000, 6, 450, 10800, 126000, 756000, 1905120, 7, 630, 18900, 294000, 2646000, 13335840, 29635200, 8, 840, 30240, 588000, 7056000, 53343360, 237081600, 479001600, 9, 1080, 45360, 1058400, 15876000, 160030080, 1066867200, 4311014400, 8083152000

Offset: 1

Jeremy Dover, Sep 25 2017

			  n=1 => AAAA -> T(1,4)=1
  n=2 => AAAA,BBBB -> T(2,4)=2
         AAAAB,AAABA,AABAA,ABAAA,BAAAA,BBBBA,BBBAB,BBABB,BABBB,ABBBB,
         AAABB,AABAB,AABBA,ABAAB,ABABA,ABBAA,BAAAB,BAABA,BABAA,BBAAA,
         BBBAA,BBABA,BBAAB,BABBA,BABAB,BAABB,ABBBA,ABBAB,ABABB,AABBB -> T(2,5)=30
Triangle begins:
1;
2,   30;
3,   90,   540;
4,  180,  2160,    8400;
5,  300,  5400,   42000,   126000;
6,  450, 10800,  126000,   756000,   1905120;
7,  630, 18900,  294000,  2646000,  13335840,   29635200;
8,  840, 30240,  588000,  7056000,  53343360,  237081600,  479001600;
...

Main diagonal is A037961.

[binomial(k,4)+10*binomial(k,5)+15*binomial(k,6)]*n!/(n+3-k)!

A292878 Number of ascending ballistic random walks of length n in 3-dimensions.

Original entry on oeis.org

1, 5, 21, 81, 313, 1213, 4701, 18217, 70593, 273557, 1060069, 4107905, 15918665, 61686893, 239044717, 926329305, 3589646289, 13910345285, 53904393461, 208886521137, 809462381657, 3136771792413, 12155397830269, 47103744291977, 182533122922465, 707339543058421, 2741032537895173, 10621856854367201

Offset: 0

Jeremy Dover, Sep 25 2017

nonn

A walk begins at the origin, and each step can be in one of five directions: Up (0,0,1), Left (-1,0,0), Right (1,0,0), Forward (0,-1,0) or Backward (0,1,0), satisfying the condition that within each plane z=k, the path may only move away from the first step into that plane. This concept generalizes the "Number of n step one-sided prudent walks with east, west and north steps" of A001333.

Number of length n strings of the symbols U, L, R, F and B such that between any L and R (resp. F and B) there appears at least one U.

			One can think of the Us as separators. Each substring between the Us (plus those before the first U and after the last U) can only contain one of L or R, and one of F or B.
Sample of a good walk: LBLL U LFFFF U U BBBRBR U FR
Sample of a bad walk: LBR U FFFRF U LBLL U FB
                      ^ ^                  ^^

Jeremy Dover, Generalization of an up, left, right path problem
Index entries for linear recurrences with constant coefficients, signature (4,-1,2).

Cf. A001333.

LinearRecurrence[{4, -1, 2}, {1, 5, 21}, 40] (* Jean-François Alcover, Sep 29 2019 *)

Vec((1 + x + 2*x^2)/(1 - 4*x + x^2 - 2*x^3) + O(x^40)) \\ Andrew Howroyd, Feb 17 2018

a(n) = a(n-1) + 4(2^n-1) + 4 Sum_{k=2..n} (2^(k-1)-1)*a(n-k) for n > 0.
From Jay Pantone, Sep 24 2017: (Start)
a(n) = 4a(n-1) - a(n-2) + 2a(n-3).
G.f.: (1 + x + 2x^2) / (1 - 4x + x^2 - 2x^3).
(End)

A281944 Triangle read by rows: T(n,k) (n>=1, 3<=k<=n+2) is the number of k-sequences of balls colored with n colors such that exactly two balls are of a color seen previously in the sequence.

Original entry on oeis.org

1, 2, 14, 3, 42, 150, 4, 84, 600, 1560, 5, 140, 1500, 7800, 16800, 6, 210, 3000, 23400, 100800, 191520, 7, 294, 5250, 54600, 352800, 1340640, 2328480, 8, 392, 8400, 109200, 940800, 5362560, 18627840, 30240000, 9, 504, 12600, 196560, 2116800, 16087680, 83825280, 272160000, 419126400, 10, 630, 18000, 327600, 4233600, 40219200, 279417600, 1360800000, 4191264000, 6187104000

Offset: 1

Jeremy Dover, Feb 02 2017

			n=1 => AAA -> T(1,3)=1
n=2 => AAA,BBB -> T(2,3)=2
   AAAB,AABA,ABAA,BAAA,BBBA,BBAB,BABB,ABBB,AABB,ABAB,ABBA,BAAB,BABA,BBAA -> T(2,4)=14
Triangle starts:
   1
   2,  14
   3,  42,   150
   4,  84,   600,   1560
   5, 140,  1500,   7800,   16800
   6, 210,  3000,  23400,  100800,   191520
   7, 294,  5250,  54600,  352800,  1340640,  2328480
   8, 392,  8400, 109200,  940800,  5362560, 18627840,  30240000
   9, 504, 12600, 196560, 2116800, 16087680, 83825280, 272160000, 419126400

Jeremy Dover, Table of n, a(n) for n = 1..1035
Jeremy Dover, Answer to Cumulative Distribution Function of Collision Counts

Columns of table: T(n,3) = A000027(n), T(n,4) = A163756(n).

Other sequences in table: T(n,n+2) = A037960(n).

Table[(Binomial[k, 3] + 3 Binomial[k, 4]) n!/(n + 2 - k)!, {n, 12}, {k, 3, n + 2}] // Flatten (* Michael De Vlieger, Feb 05 2017 *)

T(n, k) = (binomial(k,3) + 3*binomial(k,4)) * n! / (n+2-k)!;
tabl(nn) = for (n=1, nn, for (k=3, n+2, print1(T(n,k), ", ")); print()); \\ Michel Marcus, Feb 04 2017

T(n, k) = (binomial(k,3) + 3*binomial(k,4)) * n! / (n+2-k)!.

T(n, k) = n*T(n-1,k-1) + (k-2)*A281881(n,k-1).

A281946 Number of sequences of balls colored with at most n colors such that exactly two balls are of a color seen earlier in the sequence.

Original entry on oeis.org

1, 16, 195, 2248, 26245, 318936, 4082071, 55289200, 793525833, 12063384640, 194002619371, 3294811981176, 58980720557005, 1110692723476168, 21960340413007935, 455018383693865056, 9862401602086024081, 223233406292824965360, 5268151612376938762003, 129425572759622914323880

Offset: 1

Jeremy Dover, Feb 02 2017

nonn

Note that any such sequence has at least 3 balls and at most n+2.

Jeremy Dover, Table of n, a(n) for n = 1..100

Row sums of triangle A281944.

Table[n! * Sum[(Binomial[k,3]+3*Binomial[k,4])/(n+2-k)!, {k, 3, n+2}], {n, 1, 20}] (* Vaclav Kotesovec, Feb 03 2017 *)

a(n) = n! * Sum_{k=3..n+2} (binomial(k,3)+3*binomial(k,4))/(n+2-k)!.

a(n)/n! ~ e*n^4/8. - Vaclav Kotesovec, Feb 03 2017

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.

Original entry on oeis.org

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)!.

A281912 Number of sequences of balls colored with at most n colors such that exactly one ball is of a color seen earlier in the sequence.

Original entry on oeis.org

1, 8, 57, 424, 3425, 30336, 294553, 3123632, 36003969, 448816600, 6022033721, 86587079448, 1328753602657, 21683227579664, 375013198304025, 6853321766162656, 131976208783240193, 2671430511854158632, 56709161712552286009, 1259836187316759240200

Offset: 1

Jeremy Dover, Feb 01 2017

nonn

Note that any such sequence has at least 2 balls, and at most n+1

Number of 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 -> a(1) = 1.
n=2 => AA,BB,AAB,ABA,BAA,BBA,BAB,ABB -> a(2) = 8.

Jeremy Dover, Table of n, a(n) for n = 1..99

Cf. A093964.

Row sums of triangle A281881. - Jeremy Dover, Sep 26 2017

a:= proc(n) option remember;
      `if`(n<2, 1, a(n-1)*(n+2)/(n-1)-a(n-2))*n
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Feb 02 2017

Table[n!*Sum[Binomial[k, 2]/(n + 1 - k)!, {k, 2, n + 1}], {n, 20}] (* Michael De Vlieger, Feb 02 2017 *)

a(n) = n! * Sum_{k=2..n+1} binomial(k,2)/(n+1-k)!.
a(n) = n if n < 2, a(n) = n*((n+2)/(n-1)*a(n-1) - a(n-2)) for n >= 2. - Alois P. Heinz, Feb 02 2017
a(n)/n! ~ e*n^2/2. - Vaclav Kotesovec, Feb 03 2017

cp's OEIS Frontend

User: Jeremy Dover

A292999 Triangle read by rows: T(n,k) (n >= 1, 4 <= k <= n+3) is the number of k-sequences of balls colored with at most n colors such that exactly four balls are the same color as some other ball in the sequence.

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A292998 Number of sequences of balls colored with at most n colors such that exactly three balls are the same color as some other ball in the sequence.

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A292930 Triangle read by rows: T(n,k) (n>=1, 3<=k<=n+2) is the number of k-sequences of balls colored with at most n colors such that exactly three balls are the same color as some other ball in the sequence.

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A292880 Number of sequences of balls colored with at most n colors such that exactly three balls are of a color seen earlier in the sequence.

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A292879 Triangle read by rows: T(n,k) (n>=1, 4<=k<=n+3) is the number of k-sequences of balls colored with at most n colors such that exactly three balls are of a color seen previously in the sequence.

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A292878 Number of ascending ballistic random walks of length n in 3-dimensions.

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A281944 Triangle read by rows: T(n,k) (n>=1, 3<=k<=n+2) is the number of k-sequences of balls colored with n colors such that exactly two balls are of a color seen previously in the sequence.

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A281946 Number of sequences of balls colored with at most n colors such that exactly two balls are of a color seen earlier in the sequence.

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