A228273 -id:A228273 - cp's OEIS frontend

A228617 T(n,k) is the number of s in {1,...,n}^n having shortest run with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 2, 0, 24, 0, 3, 0, 240, 12, 0, 4, 0, 3080, 40, 0, 0, 5, 0, 46410, 210, 30, 0, 0, 6, 0, 822612, 840, 84, 0, 0, 0, 7, 0, 16771832, 5208, 112, 56, 0, 0, 0, 8, 0, 387395856, 23760, 720, 144, 0, 0, 0, 0, 9, 0, 9999848700, 148410, 2610, 180, 90, 0, 0, 0, 0, 10

Offset: 0

Alois P. Heinz, Aug 27 2013

Sum_{k=0..n} k*T(n,k) = A228618(n).
Sum_{k=0..n}   T(n,k) = A000312(n).
T(2*n,n)   = A002939(n) for n>0.
T(2*n+1,n) = A033586(n) for n>1.
T(2*n+2,n) = A085250(n+1) for n>2.
T(2*n+3,n) = A033586(n+1) for n>3.

			T(3,1) = 24: [1,1,2], [1,1,3], [1,2,1], [1,2,2], [1,2,3], [1,3,1], [1,3,2], [1,3,3], [2,1,1], [2,1,2], [2,1,3], [2,2,1], [2,2,3], [2,3,1], [2,3,2], [2,3,3], [3,1,1], [3,1,2], [3,1,3], [3,2,1], [3,2,2], [3,2,3], [3,3,1], [3,3,2].
T(3,3) =  3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
  1;
  0,        1;
  0,        2,    2;
  0,       24,    0,   3;
  0,      240,   12,   0,  4;
  0,     3080,   40,   0,  0,  5;
  0,    46410,  210,  30,  0,  0,  6;
  0,   822612,  840,  84,  0,  0,  0,  7;
  0, 16771832, 5208, 112, 56,  0,  0,  0,  8;

Alois P. Heinz, Rows n = 0..140, flattened

Row sums give: A000312.
Columns k=0-10 give: A000007, A228619, A228621, A228622, A228630, A228631, A228632, A228633, A228634, A228635, A228636.
Main diagonal gives: A028310.
Cf. A228154, A228273.

A085537 a(n) = n^4 - n^3.

Original entry on oeis.org

0, 0, 8, 54, 192, 500, 1080, 2058, 3584, 5832, 9000, 13310, 19008, 26364, 35672, 47250, 61440, 78608, 99144, 123462, 152000, 185220, 223608, 267674, 317952, 375000, 439400, 511758, 592704, 682892, 783000, 893730, 1015808, 1149984, 1297032, 1457750, 1632960

Offset: 0

N. J. A. Sloane, Jul 05 2003

For n>=1, a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n} such that for a fixed x in {1,2,3,4} and a fixed y in {1,2,...,n} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007
Let K_n denote the complete graph on n (n>1) vertices. The sequence corresponds to the Wiener index of K_n X K_n (Cartesian product of K_n with itself). - K.V.Iyer, Mar 12 2009
Lewis proved that the order of a solvable nonabelian finite group |G| is less than or equal to e^4 - e^3, where when d is an irreducible character degree of G, then there is a positive integer e such that |G| = d(d+e). - Jonathan Vos Post, Jun 21 2012

Michael B. Porter, Table of n, a(n) for n = 0..100000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Mark L. Lewis, Bounding group orders by large character degrees: A question of Snyder, arXiv:1206.4334 [math.GR], Jun 19 2012.
Eric Weisstein's World of Mathematics, Rook Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

A diagonal of A228273.
Cf. A000578, A000583, A092364, A146489.
Cf. A085540 (same sequence with initial 0 dropped).

Table[(n - 1) n^3, {n, 0, 20}] (* Eric W. Weisstein, Sep 08 2017 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 54, 192, 500}, {0, 20}] (* Eric W. Weisstein, Sep 08 2017 *)
CoefficientList[Series[2 x^2 (4 + 7 x + x^2)/(1 - x)^5, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)

PARI
```
A085537(n) = n^4-n^3
```

From R. J. Mathar, Sep 12 2008: (Start)
a(n) = A085540(n-1).
G.f.: 2*x^2*(4 + 7*x + x^2)/(1-x)^5. (End)
a(n) = A000583(n) - A000578(n). - Omar E. Pol, Jun 23 2012
Sum_{n>=2} 1/a(n) = 3 - zeta(2) - zeta(3) = A152419. - Daniel Suteu, Feb 06 2017
a(n) = 2*A092364(n+1). - Bruno Berselli, Sep 08 2017
Sum_{n>=2} (-1)^n/a(n) = Pi^2/12 + 2*log(2) + 3*zeta(3)/4 - 3. - Amiram Eldar, Jul 05 2020
E.g.f.: exp(x)*x^2*(4 + 5*x + x^2). - Stefano Spezia, Jul 06 2021
Product_{n>=2} (1 - 1/a(n)) = A146489. - Amiram Eldar, Nov 22 2022

A031972 a(n) = Sum_{k=1..n} n^k.

Original entry on oeis.org

0, 1, 6, 39, 340, 3905, 55986, 960799, 19173960, 435848049, 11111111110, 313842837671, 9726655034460, 328114698808273, 11966776581370170, 469172025408063615, 19676527011956855056, 878942778254232811937, 41660902667961039785742, 2088331858752553232964199

Offset: 0

N. J. A. Sloane, Dec 11 1999

Sum of lengths of longest ending contiguous subsequences with the same value over all s in {1,...,n}^n: a(n) = Sum_{k=1..n} k*A228273(n,k). a(2) = 6 = 2+1+1+2: [1,1], [1,2], [2,1], [2,2]. - Alois P. Heinz, Aug 19 2013
a(n) is the expected wait time to see the contiguous subword 11...1 (n copies of 1) over all infinite sequences on alphabet {1,2,...,n}. - Geoffrey Critzer, May 19 2014
a(n) is the number of sequences of k elements from {1,2,...,n}, where 1<=k<=n. For example, a(2) = 6, counting the sequences, [1], [2], [1,1], [1,2], [2,1], [2,2]. Equivalently, a(n) is the number of bar graphs having a height and width of at most n. - Emeric Deutsch, Jan 24 2017.
In base n, a(n) has n+1 digits: n 1's followed by a 0. - Mathew Englander, Oct 20 2020

Alois P. Heinz, Table of n, a(n) for n = 0..386
A. Blecher, C. Brennan, A. Knopfmacher and H. Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44.

Main diagonal of A228275.

Cf. A031973, A228273, A023037, A226238.

a031972 n = sum $ take n $ iterate (* n) n
-- Reinhard Zumkeller, Nov 22 2014

[1] cat [(n^(n+1)-n)/(n-1): n in [2..20]]; // Vincenzo Librandi, Apr 16 2015

a:= n-> `if`(n<2, n, (n^(n+1)-n)/(n-1)):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 15 2013

f[n_]:=Sum[n^k,{k,n}];Array[f,30] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)

a(1)=1; for n!=1 a(n) = (n^(n+1)-1)/(n-1) - 1. - Benoit Cloitre, Aug 17 2002
a(n) = A031973(n)-1 for n>0. - Robert G. Wilson v, Apr 15 2015
a(n) = n*A023037(n) = n^n - 1 + A023037(n). - Mathew Englander, Oct 20 2020

a(0)=0 prepended by Alois P. Heinz, Oct 22 2019

A085538 a(n) = n^5 - n^4.

Original entry on oeis.org

0, 0, 16, 162, 768, 2500, 6480, 14406, 28672, 52488, 90000, 146410, 228096, 342732, 499408, 708750, 983040, 1336336, 1784592, 2345778, 3040000, 3889620, 4919376, 6156502, 7630848, 9375000, 11424400, 13817466, 16595712, 19803868, 23490000, 27705630, 32505856

Offset: 0

N. J. A. Sloane, Jul 05 2003

For n >= 1, a(n) is equal to the number of functions f:{1,2,3,4,5}->{1,2,...,n} such that for a fixed x in {1,2,3,4,5} and a fixed y in {1,2,...,n} we have f(x) <> y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

A diagonal of A228273.

Cf. A000583, A000584, A146492.

[n^5-n^4: n in [0..50]]; // Vincenzo Librandi, Feb 12 2012

a:=n->sum(sum(n^3, j=1..n),k=2..n): seq(a(n), n=0..31); # Zerinvary Lajos, May 09 2007

Table[n^5 - n^4, {n, 0, 40}] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 16, 162, 768, 2500}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)

a(n)=n^5-n^4 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 2*x^2*(x^3 + 18*x^2 + 33*x + 8)/(x-1)^6. - Colin Barker, Nov 06 2012
Sum_{n>=2} 1/a(n) = 4 - zeta(2) - zeta(3) - zeta(4). - Amiram Eldar, Jul 05 2020
Product_{n>=2} (1 - 1/a(n)) = A146492. - Amiram Eldar, Nov 22 2022

A085539 a(n) = n^6 - n^5.

Original entry on oeis.org

0, 0, 32, 486, 3072, 12500, 38880, 100842, 229376, 472392, 900000, 1610510, 2737152, 4455516, 6991712, 10631250, 15728640, 22717712, 32122656, 44569782, 60800000, 81682020, 108226272, 141599546, 183140352, 234375000, 297034400, 373071582, 464679936

Offset: 0

N. J. A. Sloane, Jul 05 2003

For n>=1, a(n) is equal to the number of functions f:{1,2,3,4,5,6}->{1,2,...,n} such that for a fixed x in {1,2,3,4,5,6} and a fixed y in {1,2,...,n} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

A diagonal of A228273.

Cf. A000584, A001014.

[n^6-n^5: n in [0..40]]; // Vincenzo Librandi, Aug 15 2016

f[n_]:=n^6-n^5;f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)
Table[n^6 - n^5, {n, 0, 50}] (* Vincenzo Librandi, Aug 15 2016 *)

a(n)=n^6-n^5 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: -2*x^2*(x^4+41*x^3+171*x^2+131*x+16)/(x-1)^7. - Colin Barker, Nov 06 2012

Sum_{n>=2} 1/a(n) = 5 - Sum_{k=2..5} zeta(k). - Amiram Eldar, Jul 05 2020

A228154 T(n,k) is the number of s in {1,...,n}^n having longest contiguous subsequence with the same value of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 2, 2, 12, 12, 3, 108, 120, 24, 4, 1280, 1520, 280, 40, 5, 18750, 23400, 3930, 510, 60, 6, 326592, 423360, 65016, 7644, 840, 84, 7, 6588344, 8800008, 1241464, 132552, 13440, 1288, 112, 8, 150994944, 206622720, 26911296, 2622528, 244944, 22032, 1872, 144, 9

Offset: 1

Walt Rorie-Baety, Aug 15 2013

			T(1,1) =  1: [1].
T(2,1) =  2: [1,2], [2,1].
T(2,2) =  2: [1,1], [2,2].
T(3,1) = 12: [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3].
T(3,2) = 12: [1,1,2], [1,1,3], [1,2,2], [1,3,3], [2,1,1], [2,2,1], [2,2,3], [2,3,3], [3,1,1], [3,2,2], [3,3,1], [3,3,2].
T(3,3) =  3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
.       1;
.       2,       2;
.      12,      12,       3;
.     108,     120,      24,      4;
.    1280,    1520,     280,     40,     5;
.   18750,   23400,    3930,    510,    60,    6;
.  326592,  423360,   65016,   7644,   840,   84,   7;
. 6588344, 8800008, 1241464, 132552, 13440, 1288, 112,  8;

Alois P. Heinz, Rows n = 1..141, flattened
Project Euler, Problem 427: n-sequences

Row sums give: A000312.
Column k=1 gives: A055897.
Main diagonal gives: A000027.
Lower diagonal gives: 2*A180291.
Cf. A228194, A228273, A228617.

T:= proc(n) option remember; local b; b:=
      proc(m, s, i) option remember; `if`(m>i or s>m, 0,
        `if`(i=1, n, `if`(s=1, (n-1)*add(b(m, h, i-1), h=1..m),
         b(m, s-1, i-1) +`if`(s=m, b(m-1, s-1, i-1), 0))))
      end; forget(b);
      seq(add(b(k, s, n), s=1..k), k=1..n)
    end:
seq(T(n), n=1..12);  # Alois P. Heinz, Aug 18 2013

T[n_] := T[n] = Module[{b}, b[m_, s_, i_] := b[m, s, i] = If[m>i || s>m, 0, If[i == 1, n, If[s == 1, (n-1)*Sum[b[m, h, i-1], {h, 1, m}], b[m, s-1, i-1] + If[s == m, b[m-1, s-1, i-1], 0]]]]; Table[Sum[b[k, s, n], {s, 1, k}], {k, 1, n}]]; Table[ T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Mar 06 2015, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A228194(n). - Alois P. Heinz, Dec 23 2020

cp's OEIS Frontend

A228617 T(n,k) is the number of s in {1,...,n}^n having shortest run with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A085537 a(n) = n^4 - n^3.

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A031972 a(n) = Sum_{k=1..n} n^k.

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A085538 a(n) = n^5 - n^4.

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A085539 a(n) = n^6 - n^5.

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A228154 T(n,k) is the number of s in {1,...,n}^n having longest contiguous subsequence with the same value of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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