A000255 -id:A000255 - cp's OEIS frontend

A010029 Irregular triangle read by rows: T(n,k) (n>=1, 0 <= k <= floor(n/2)) = number of permutations of 1..n with exactly floor(n/2) - k runs of consecutive pairs up.

1, 1, 1, 3, 3, 1, 12, 11, 11, 56, 53, 3, 87, 321, 309, 53, 693, 2175, 2119, 11, 680, 5934, 17008, 16687, 309, 8064, 55674, 150504, 148329, 53, 5805, 96370, 572650, 1485465, 1468457, 2119, 95575

Offset: 1

N. J. A. Sloane

			Triangle begins
     1
     1     1
     3     3
     1    12    11
    11    56    53
     3    87   321    309
    53   693  2175   2119
    11   680  5934  17008   16687
   309  8064 55674 150504  148329
    53  5805 96370 572650 1485465 1468457
  2119 95575 ...
  ...

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264, Table 7.6.1.

Vincenzo Librandi, Rows n = 1..50, flattened

Cf. A000255, A001277, A001278, A001279, A001280, A000142 (row sums), A136123 (rows reversed).

A010029 := proc(n,k)
    add( x^i*( ((1-y)*x-1)/((1-y)*x^2-1) )^i*i!,i=0..n+1) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,floor(n/2)-k) ;
end proc:
seq(seq( A010029(n,k),k=0..floor(n/2)),n=1..12) ; # R. J. Mathar, Jul 01 2022

max = 16; coes = CoefficientList[ Series[ Sum[ n!*(((1 - y)*x^2 - x)/((1 - y)*x^2 - 1))^n, {n, 0, max}], {x, 0, max}, {y, 0, max}], {x, y}]; Table[ Table[ coes[[n, k]] , {k, 1, Floor[(n + 1)/2]}] // Reverse, {n, 2, max - 4}] // Flatten (* Jean-François Alcover, Jan 10 2013, after Vladeta Jovovic *)

G.f.: Sum_{n>=0} n! *( ((1-y)*x^2-x)/((1-y)*x^2-1) )^n, for the triangle read right-to-left. - Vladeta Jovovic, Nov 21 2007

T(n,k) = A136123(n,[n/2]-k). - R. J. Mathar, Jul 01 2022

A046740 Triangle of number of permutations of [n] with 0 successions, by number of rises.

Original entry on oeis.org

1, 1, 1, 2, 1, 8, 2, 1, 22, 28, 2, 1, 52, 182, 72, 2, 1, 114, 864, 974, 164, 2, 1, 240, 3474, 8444, 4174, 352, 2, 1, 494, 12660, 57194, 61464, 15782, 732, 2, 1, 1004, 43358, 332528, 660842, 373940, 55286, 1496, 2, 1, 2026, 142552, 1747558, 5814124

Offset: 1

N. J. A. Sloane

The recurrence given by Roselle is wrong.

			Triangle begins:
  1;
  1;
  1,  2;
  1,  8,  2;
  1, 22, 28,  2;
  ...

D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16.
D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16. [Annotated scanned copy]

Cf. A046739, A000295. Row sums give A000255. Diagonals give A005803, A065340.

Row sums give A000255.

a[, 1] = 1; a[n, 2] := 2^n - 2*n; a[n_, r_] /; 1 <= r <= n-1 := a[n, r] = r*a[n-1, r] + (n-r)*a[n-1, r-1] + (n-2)*a[n-2, r-1]; a[, ] = 0;
row[1] = {{1}}; row[n_] := Table[a[n, r], {r, 1, n-1}];
Table[row[n], {n, 1, 11}] // Flatten (* Jean-François Alcover, Sep 07 2017 *)

a(n, 1) = 1; for r > 1, a(n, r) = r*a(n-1, r) + (n-r)*a(n-1, r-1) + (n-2)*a(n-2, r-1).

a(n, 2) = 2^n - 2*n = 2*A000295 = A005803, n >= 3.

More terms from Vladeta Jovovic, Jan 03 2003

A090013 Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.

Original entry on oeis.org

4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284, 514872176, 7058605844, 103969203576, 1637182717924, 27442553929696, 487806792137844, 9164718013496936, 181446744138509444, 3775570370986139856

Offset: 1

Jaap Spies, Dec 13 2003

Brualdi, Richard A. and Ryser, Herbert J., Combinatorial Matrix Theory, Cambridge NY (1991), Chapter 7.

Indranil Ghosh, Table of n, a(n) for n = 1..446
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), pp. 197-210.

a(n) = A000261(n-1) + A000261(n), a(1)=4

Cf. A000255, A000153, A000261, A001909, A001910, A090010, A055790, A090012-A090016.

t={4,16};Do[AppendTo[t,(n+2)*t[[-1]]+(n-2)*t[[-2]]],{n,3,18}];t (* Indranil Ghosh, Feb 21 2017 *)

a(n) = (n+2)*a(n-1) + (n-2)*a(n-2), a(1)=4, a(2)=16

a(n) ~ exp(-1) * n! * n^3 / 6. - Vaclav Kotesovec, Nov 30 2017

Corrected by Jaap Spies, Jan 26 2004

A090015 Permanent of (0,1)-matrix of size n X (n+d) with d=5 and n-1 zeros not on a line.

Original entry on oeis.org

6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774, 5930064372, 92608986546, 1541044428456, 27216454135758, 508388707585116, 10013199347882058, 207381428863832784, 4505207996358719334

Offset: 1

Jaap Spies, Dec 13 2003

Brualdi, Richard A. and Ryser, Herbert J., Combinatorial Matrix Theory, Cambridge NY (1991), Chapter 7.

Indranil Ghosh, Table of n, a(n) for n = 1..445
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), pp. 197-210.

a(n) = A001910(n-1) + A001910(n), a(1)=6

Cf. A000255, A000153, A000261, A001909, A001910, A090010, A055790, A090012-A090016.

f:= gfun:-rectoproc({a(n) = (n+4)*a(n-1) + (n-2)*a(n-2),a(1)=6,a(2)=36},a(n),remember):
map(f, [$1..40]); # Robert Israel, Nov 26 2018

t={6,36};Do[AppendTo[t,(n+4)*t[[-1]]+(n-2)*t[[-2]]],{n,3,17}];t (* Indranil Ghosh, Feb 21 2017 *)
RecurrenceTable[{a[n] == (n+4)*a[n-1] + (n-2)*a[n-2],
a[1] == 6, a[2] == 36}, a, {n, 1, 40}] (* Jean-François Alcover, Sep 16 2022, after Robert Israel *)

a(n) = (n+4)*a(n-1) + (n-2)*a(n-2), a(1)=6, a(2)=36
a(n) ~ exp(-1) * n! * n^5 / 5!. - Vaclav Kotesovec, Nov 30 2017
a(n) = ((n^6+21*n^5+160*n^4+545*n^3+814*n^2+415*n+1)*exp(-1)*Gamma(n, -1)+(-1)^n*(n^5+20*n^4+141*n^3+422*n^2+499*n+154))/120. - Robert Israel, Nov 26 2018

Corrected by Jaap Spies, Jan 26 2004

A113059 a(n) = n! * Sum_{k=0..n} A000296(k)/k!.

Original entry on oeis.org

1, 1, 3, 10, 44, 231, 1427, 10151, 81923, 740732, 7425042, 81773715, 981864897, 12767876941, 178774288331, 2681781213130, 42909715480460, 729474427239587, 13130613291110603, 249482261007109579, 4989650444408388515, 104782705832468197252, 2305219956684224457858

Offset: 0

Karol A. Penson, Oct 12 2005

nonn

Number of set partitions of [n] where the k-th singletons are k-colored and all other blocks are unicolored. - Alois P. Heinz, Apr 29 2025

G. C. Greubel, Table of n, a(n) for n = 0..448

Cf. A000110, A000142, A000255, A000296, A101053, A101054.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(Exp(x)-1-x)/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 23 2018

b:= proc(n, k) option remember; `if`(n=0, 1, k*b(n-1, k+1)+
      add(b(n-j, k)*binomial(n-1, j-1), j=2..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 29 2025
# second Maple program:
b:= proc(n, k, m) option remember; `if`(n=0, k!, `if`(k>0,
      b(n-1, k-1, m+1)*k, 0)+m*b(n-1, k, m)+b(n-1, k+1, m))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 29 2025

With[{nmax = 50}, CoefficientList[Series[Exp[Exp[x] - 1 - x]/(1 - x), {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, May 23 2018 *)

x='x+O('x^30); Vec(serlaplace( exp(exp(x)-1-x)/(1-x))) \\ G. C. Greubel, May 23 2018

a(n) = (-1)^n*n!*Sum_{k >=0} LaguerreL(n, -n-1, k-1)/k!/exp(1), n>=0.
E.g.f.: exp(exp(x)-1-x)/(1-x).
a(n) ~ exp(exp(1)-2) * n!. - Vaclav Kotesovec, Jun 26 2022

A153869 Triangle read by rows, A129186 * A128064(unsigned).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 0, 3, 4, 0, 0, 0, 0, 4, 5, 0, 0, 0, 0, 0, 5, 6, 0, 0, 0, 0, 0, 0, 6, 7, 0, 0, 0, 0, 0, 0, 0, 7, 8, 0, 0, 0, 0, 0, 0, 0, 0, 8, 9, 0

Offset: 1

Gary W. Adamson, Jan 03 2009

Lim_{k->inf} A153869^n = A000255: (1, 1, 3, 11, 53, 309, 2119,...).
Row sums = (1, 1, 3, 5, 7, 9,...).
A153869 * (1, 2, 3,...) = A001844 prefaced with a 1: (1, 1, 5, 13, 25, 41,...).

			First few rows of the triangle =
1;
1, 0;
1, 2, 0;
0, 2, 3, 0;
0, 0, 3, 4, 0;
0, 0, 0, 4, 5, 0;
0, 0, 0, 0, 5, 6, 0;
0, 0, 0, 0, 0, 6, 7, 0;
0, 0, 0, 0, 0, 0, 7, 8, 0;
...

Cf. A128064, A129186, A000255, A001844

Triangle read by rows, A129186 * A128064; where A129186 = a shift operator, shifting down triangle A128064(unsigned) one row and inserting a "1" at (1,1).

A174549 a(n) = (2n-1)! + (2n)!.

Original entry on oeis.org

3, 30, 840, 45360, 3991680, 518918400, 93405312000, 22230464256000, 6758061133824000, 2554547108585472000, 1175091669949317120000, 646300418472124416000000, 418802671169936621568000000, 315777214062132212662272000000, 274094621805930760590852096000000

Offset: 1

Paul Curtz, Mar 22 2010

x*cos(x) - sin(x) = Sum_{n>=1} (-1)^n/a(n) * x^(2*n+1). - James R. Buddenhagen, Nov 21 2013

Also the number of adjacency matrices for the n-helm graph. - Eric W. Weisstein, May 25 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Adjacency Matrix.
Eric Weisstein's World of Mathematics, Helm Graph.

Cf. A001048, A000255, A068985, A143624, A165457.

[Factorial(2*n-1) + Factorial(2*n): n in [1..15]]; // Vincenzo Librandi, Aug 04 2011

A174549 := proc(n) (1+2*n)*(2*n-1)! ; end proc: # R. J. Mathar, Jan 13 2011

Table[(2*n - 1)! + (2*n)!, {n, 15}] (* T. D. Noe, Nov 21 2013 *)

a(n)=(2*n-1)!*(2*n+1) \\ Charles R Greathouse IV, Nov 21 2013

a(n) = A001048(2n) = (1+2n)*(2n-1)! = 3*A165457(n-1).
Sum_{n>=1} 1/a(n) = A068985 = 1/e = lim_{n->infinity} A000255(n-1)/A001048(n).
zeta(2*n+1) = Integral_{u=0..Pi/2} (sin(u)*log(sin(u))^(2*n+1)/(cos(u)^3))*(-2)^(2*n+1)/(n*a(n)) du. Verified for n=1 to 4 on Wolfram Alpha. - Jean-Claude Babois, Oct 28 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = sin(1)-cos(1) = (-1)*A143624. - Amiram Eldar, Apr 12 2021

A180196 Triangle read by rows: T(n,k) is the number of permutations of [n] that have k isolated entries (0 <= k <= n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 0, 3, 2, 2, 9, 0, 11, 3, 11, 9, 44, 0, 53, 7, 20, 75, 44, 265, 0, 309, 14, 73, 141, 574, 265, 1854, 0, 2119, 35, 170, 737, 1104, 4900, 1854, 14833, 0, 16687, 81, 576, 1863, 7814, 9535, 46353, 14833, 133496, 0, 148329, 216, 1556, 8154, 20704, 88335, 90852, 482069, 133496, 1334961, 0, 1468457

Offset: 0

Emeric Deutsch, Sep 09 2010

An entry j of a permutation p is isolated if it is not preceded by j-1 and not followed by j+1. For example, the permutation 23178564 has 2 isolated entries: 1 and 4.
Sum of entries in row n is n! = A000142(n).
T(n,n) = d(n) + d(n-1) = A000255(n-1), where d(i)=A000166(i) are the derangement numbers.
T(n,n-2) = d(n) (n >= 2).
T(n,n-3) = d(n-1) (n >= 3).
Sum_{k=0..n} k*T(n,k) = (n-2)!*(n^3 - 3n^2 + 5n - 4) = A001565(n-2) (n >= 2).

			T(4,2)=9 because we have 124'3', 1'4'23, 1'342', 3'124', 4'3'12, 2'1'34, 231'4', 4'231', and 342'1' (the isolated entries are marked).
Triangle starts:
  1;
  0,  1;
  1,  0,  1;
  1,  2,  0,  3;
  2,  2,  9,  0, 11;
  3, 11,  9, 44,  0, 53;

Cf. A000142, A000166, A000255, A001565.

d[ -1] := 0: d[0] := 1: for n to 50 do d[n] := n*d[n-1]+(-1)^n end do: T := proc (n, k) if k < n then sum(binomial(n-1-j, j-k-1)*binomial(j, k)*(d[j]+d[j-1]), j = k+1 .. floor((1/2)*n+(1/2)*k)) elif k = n then d[n]+d[n-1] else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form

T[n_, k_] := With[{d = Subfactorial}, Which[k == n == 0, 1, k == n, d[n] + d[n - 1], True, Sum[Binomial[n - 1 - j, j - k - 1]*Binomial[j, k]*(d[j] + d[j - 1]), {j, k + 1, Floor[(n + k)/2]}]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2024 *)

T(n,k) = Sum_{j=k+1..floor((n+k)/2)} binomial(n-1-j, j-k-1)*binomial(j,k)*(d(j) + d(j-1)), if k < n;

T(n,n) = d(n) + d(n-1); d(i)=A000166(i) are the derangement numbers.

A189284 Number of permutations p of 1,2,...,n satisfying p(i+5)-p(i)<>5 for all 1<=i<=n-5.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 696, 4572, 34260, 290328, 2751480, 28686024, 328764732, 4106158164, 55495145304, 806797105320, 12554890849992, 208164423163908, 3663256621120548, 68188490015132040, 1338490745511631080, 27630826605742438968

Offset: 0

Vaclav Kotesovec, Apr 19 2011

a(n) is also number of ways to place n nonattacking pieces rook + semi-leaper[5,5] on an n X n chessboard.

Vaclav Kotesovec, Table of n, a(n) for n = 0..26 (Updated Jan 19 2019)
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 644.
Vaclav Kotesovec, Mathematica program for this sequence
George Spahn and Doron Zeilberger, Counting Permutations Where The Difference Between Entries Located r Places Apart Can never be s (For any given positive integers r and s), arXiv:2211.02550 [math.CO], 2022.

Cf. A000255, A189281, A189282, A189283, A189256.

Asymptotics (V. Kotesovec, Mar 2011): a(n)/n! ~ (1 + 9/n + 20/n^2)/e.

Terms a(25)-a(26) from Vaclav Kotesovec, Apr 20 2012

A269954 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 2, 5, 3, 1, 0, 9, 20, 17, 6, 1, 0, 44, 109, 100, 45, 10, 1, 0, 265, 689, 694, 355, 100, 15, 1, 0, 1854, 5053, 5453, 3094, 1015, 196, 21, 1, 0, 14833, 42048, 48082, 29596, 10899, 2492, 350, 28, 1

Offset: 0

Peter Luschny, Apr 12 2016

			Triangle starts:
  1;
  0,   1;
  0,   0,   1;
  0,   1,   1,   1;
  0,   2,   5,   3,   1;
  0,   9,  20,  17,   6,   1;
  0,  44, 109, 100,  45,  10,  1;
  0, 265, 689, 694, 355, 100, 15, 1;

Peter Luschny, Extensions of the binomial

A000255 (row sums), A000217 (diag. n,n-1), A133252 (diag. n,n-2).
Columns k=0..4 give A000007, A000166(n-1), A300490(n-1), A381067(n-1), A381068(n-1).
Cf. A269951, A269953.

A269954 := (n, k) -> add(binomial(-j, -n)*abs(Stirling1(j, k)), j=0..n):
seq(seq(A269954(n, k), k=0..n), n=0..9);

Flatten[Table[Sum[Binomial[-j,-n] Abs[StirlingS1[j,k]],{j,0,n}], {n,0,9},{k,0,n}]]

T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(n-1, n-j)*abs(stirling(j, k)));
for(n=0, 9, for(k=0, n, print1(T(n, k), ", "))); \\ Seiichi Manyama, Feb 13 2025

cp's OEIS Frontend

A010029 Irregular triangle read by rows: T(n,k) (n>=1, 0 <= k <= floor(n/2)) = number of permutations of 1..n with exactly floor(n/2) - k runs of consecutive pairs up.

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A046740 Triangle of number of permutations of [n] with 0 successions, by number of rises.

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A090013 Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.

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A090015 Permanent of (0,1)-matrix of size n X (n+d) with d=5 and n-1 zeros not on a line.

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A113059 a(n) = n! * Sum_{k=0..n} A000296(k)/k!.

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A153869 Triangle read by rows, A129186 * A128064(unsigned).

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A174549 a(n) = (2*n-1)! + (2*n)!.

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A174549 a(n) = (2n-1)! + (2n)!.