A000240 -id:A000240 - cp's OEIS frontend

A161130 Sum of the differences between the largest and the smallest fixed points over all non-derangement permutations of {1,2,...,n}.

0, 0, 1, 2, 13, 74, 523, 4178, 37609, 376082, 4136911, 49642922, 645357997, 9035011946, 135525179203, 2168402867234, 36862848742993, 663531277373858, 12607094270103319, 252141885402066362, 5294979593443393621

Offset: 0

Emeric Deutsch, Jul 18 2009

nonn

			a(3)=2 because the non-derangements of {1,2,3} are 1'23', 1'32, 213', and 32'1 with differences between the largest and smallest fixed points (marked) equal to 2, 0, 0, and 0, respectively.
a(4)=13 because the non-derangements of {1,2,3,4} are 1'234', 1'2'43, 1'423, 1'324', 1'342, 1'43'2, 413'2, 3124', 213'4', 42'13, 2314', 243'1, 42'3'1, 32'14', and 32'41 with differences between the largest and smallest fixed points (marked) equal to 3, 1, 0, 3, 0, 2, 0, 0, 1, 0, 0, 0, 1, 2, and 0, respectively.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
E. Deutsch and S. Elizalde, The largest and the smallest fixed points of permutations, arXiv:0904.2792v1 [math.CO], 2009.

Cf. A000166, A000240, A155521, A161129

G := (exp(-x)*(1+x+x^2)-1)/(1-x)^2: Gser := series(G, x = 0, 25): seq(factorial(n)*coeff(Gser, x, n), n = 0 .. 22);

CoefficientList[Series[(E^(-x)*(1+x+x^2)-1)/(1-x)^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)

E.g.f.: (exp(-x) * (1+x+x^2) - 1) / (1-x)^2.
a(n) = A000166(n+1) - A155521(n).
a(n) = Sum(k*A161129(n,k), k=0..n-1).
Recurrence: (n-2)*a(n) = (n^2-2*n-1)*a(n-1) + (n-1)*n*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ n!*n*(3/e-1). - Vaclav Kotesovec, Oct 20 2012

A182390 a(0)=0, a(n) = (a(n-1) * n) XOR n.

Original entry on oeis.org

0, 1, 0, 3, 8, 45, 264, 1855, 14832, 133497, 1334960, 14684571, 176214856, 2290793125, 32071103752, 481066556279, 7697064900448, 130850103307633, 2355301859537376, 44750735331210163, 895014706624203240, 18795308839108268061, 413496794460381897320

Offset: 0

Alex Ratushnyak, Apr 27 2012

For 0A000240(n-1).

Harvey P. Dale, Table of n, a(n) for n = 0..450

Cf. A182249, A000240.

nxt[{n_,a_}]:={n+1,BitXor[a(n+1),n+1]}; NestList[nxt,{0,0},30][[;;,2]] (* Harvey P. Dale, Mar 14 2023 *)

a=0
for i in range(1,55):
    print(a, end=', ')
    a *= i
    a ^= i

a(0)=0, a(n) = (a(n-1) * n) XOR n, where XOR is the bitwise exclusive-OR operator.

A211229 Matrix inverse of lower triangular array A211226.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 0, 0, -1, 1, 1, 0, 0, -2, 1, -1, 1, 0, 0, -1, 1, 2, -3, 3, 0, 0, -3, 1, -2, 2, -3, 3, 0, 0, -1, 1, 9, -8, 8, -12, 6, 0, 0, -4, 1, -9, 9, -8, 8, -6, 6, 0, 0, -1, 1, 44, -45, 45, -40, 20, -30, 10, 0, 0, -5, 1, -44, 44, -45, 45, -20, 20, -10, 10, 0, 0, -1, 1

Offset: 0

Peter Bala, Apr 05 2012

This triangle is related to the derangement numbers. The subtriangles (T(2*n,2*k))n,k>=0, -(T(2*n+1,2*k))n,k>=0, and (T(2*n+1,2*k+1))n,k>=0 are all equal to A008290, while the subtriangle (T(2*n,2*k+1))n,k>=0 equals -A180188 (with an extra initial row of zeros).

			Triangle begins:
   n\k |    0    1    2    3    4    5    6    7    8    9
  =====+==================================================
    0  |    1
    1  |   -1    1
    2  |    0   -1    1
    3  |    0    0   -1    1
    4  |    1    0    0   -2    1
    5  |   -1    1    0    0   -1    1
    6  |    2   -3    3    0    0   -3    1
    7  |   -2    2   -3    3    0    0   -1    1
    8  |    9   -8    8  -12    6    0    0   -4    1
    9  |   -9    9   -8    8   -6    6    0    0   -1    1
  ...

Cf. A000166, A000240, A000387, A000449, A000475, A008290, A180188, A211226.

b[j_] = Floor[j/2]; h = If[EvenQ[n] && OddQ[k], 1, 0];
Table[(-1)^(n+k) (b[n]!/b[k]!) Sum[(-1)^i/i!, {i, 0, b[n-k]-h}], {n, 0, 31}, {k, 0, n}] //Flatten (* Manfred Boergens, Jan 10 2023 *)
(* Sum-free code *)
b[j_] = Floor[j/2]; h = If[EvenQ[n] && OddQ[k], 1, 0];
T[n_, k_] = (-1)^(n+k) (b[n]!/b[k]!) If[n-k<2, 1, Round[(b[n-k]-h)!/E]/(b[n-k]-h)!];
Table[T[n, k], {n, 0, 31}, {k, 0, n}] // Flatten
(* Manfred Boergens, Jan 10 2023 *)

f(n) = (n\2)!; \\ A081123
T(n,k) = f(n)/(f(k)*f(n-k)); \\ A211226
tabl(nn) = my(m=matrix(nn, nn, n, k, if (n>=k, T(n-1,k-1), 0))); 1/m; \\ Michel Marcus, Jan 10 2023

T(2*n,2*k) = T(2*n+1,2*k+1) = -T(2*n+1,2*k) = binomial(n,k)*A000166(n-k) = (n!/k!)*Sum_{i = 0..n-k} (-1)^i/i!;
T(2*n,2*k+1) = -n*binomial(n-1,k)*A000166(n-k-1) = -(n!/k!)*Sum_{i = 0..n-k-1} (-1)^i/i!.
T(n,k) = T(n-k,0)*A211226(n,k).
Column entries:
T(2*n,0) = A000166(n), T(2*n,2) = A000240(n), T(2*n,4) = A000387(n), T(2*n,6) = A000449(n), T(2*n,8) = A000475(n).
From Manfred Boergens, Jan 10 2023: (Start)
With b(j) = floor(j/2); h = 1 for n even and k odd, h = 0 else:
T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!)*Sum_{i = 0..b(n-k)-h} (-1)^i/i!.
Sum-free formula:
T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!) for n-k < 2.
T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!)*round((b(n-k)-h)!/exp(1))/(b(n-k)-h)!) otherwise. (End)

More terms from Manfred Boergens, Jan 10 2023

A235378 a(n) = (-1)^n*(n! - (-1)^n).

Original entry on oeis.org

-2, 1, -7, 23, -121, 719, -5041, 40319, -362881, 3628799, -39916801, 479001599, -6227020801, 87178291199, -1307674368001, 20922789887999, -355687428096001, 6402373705727999, -121645100408832001, 2432902008176639999, -51090942171709440001, 1124000727777607679999

Offset: 1

Jean-François Alcover, Jan 08 2014

This sequence links rencontres numbers r(n) with Sum_{k>=1} 1/((k+n)*k!) = (a(n) + (-1)^(n+1)*e*r(n))/n.

G. C. Greubel, Table of n, a(n) for n = 1..448 (terms 1..40 from Jean-François Alcover)
Fred Kline, Question asked on MathOverflow: Sum_{k=0..infinity} 1/((k+2)*k!)

Cf. A000240.

[(-1)^n*(Factorial(n) - (-1)^n): n in [1..30]]; // G. C. Greubel, Nov 21 2017

r[n_] := n*Subfactorial[n-1]; a[n_] := n*Sum[1/((k + n)*k!), {k, 1, Infinity}] + (-1)^n*E*r[n]; Table[a[n], {n, 1, 25}]
(* or, simply: *) Table[(-1)^n*(n!-(-1)^n), {n, 1, 25}]
a[1]:=-2;a[n]:=(-1)^n*Sum[Abs[StirlingS1[n,j]+Binomial[n-1,j]],{j,0,n-1}];Flatten[Table[a[n],{n,1,19}]] (* Detlef Meya, Apr 11 2024 *)

for(n=1, 30, print1((-1)^n*(n!-(-1)^n), ", ")) \\ G. C. Greubel, Nov 21 2017

Recurrence: a(1)=-2, a(2)=1; for n>2, a(n) = -n*a(n-1) - n - 1.
E.g.f.: 1/(1+x) - exp(x).
D-finite with recurrence: a(n) +(n-2)*a(n-1) +(-2*n+3)*a(n-2) +(n-2)*a(n-3)=0. - R. J. Mathar, Feb 24 2020
a(1) = -2; For a > 1: a(n) = (-1)^n*Sum_{j=0..n-1} (abs(Stirling1(n,j) + binomial(n - 1, j))). - Detlef Meya, Apr 11 2024

A318365 Expansion of e.g.f. exp(x*exp(-x)/(1 - x)).

Original entry on oeis.org

1, 1, 1, 4, 21, 116, 805, 6504, 59353, 608320, 6901641, 85824080, 1160786341, 16959401304, 266133942061, 4463567862376, 79669223849265, 1507610621184224, 30145968665822737, 635066714078714016, 14057275047440540221, 326159212986987669640, 7915118313077599105461, 200503241124736099689656

Offset: 0

Ilya Gutkovskiy, Aug 24 2018

nonn

Cf. A000166, A000240, A000262, A003725, A318364.

seq(n!*coeff(series(exp(x*exp(-x)/(1-x)),x=0,24),x,n),n=0..23); # Paolo P. Lava, Jan 09 2019

nmax = 23; CoefficientList[Series[Exp[x Exp[-x]/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[k Subfactorial[k - 1] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

x = 'x + O('x^25); Vec(serlaplace(exp(x*exp(-x)/(1 - x)))) \\ Michel Marcus, Aug 25 2018

a(0) = 1; a(n) = Sum_{k=1..n} A000240(k)*binomial(n-1,k-1)*a(n-k).

a(n) ~ exp(exp(-1)/2 - 1/4 + 2*exp(-1/2)*sqrt(n) - n) * n^(n - 1/4) / sqrt(2). - Vaclav Kotesovec, Aug 25 2018

A331796 E.g.f.: (exp(x) - 1) * exp(1 - exp(x)) / (2 - exp(x)).

Original entry on oeis.org

0, 1, 1, 4, 27, 201, 1730, 17403, 200753, 2607034, 37614509, 596935373, 10334325760, 193820393781, 3914731176005, 84716449797164, 1955520065429447, 47960724916860501, 1245468600257306394, 34139796085144434199, 985066290121984334613

Offset: 0

Ilya Gutkovskiy, Jan 26 2020

nonn

Stirling transform of A000240.

Alois P. Heinz, Table of n, a(n) for n = 0..424

Cf. A000240, A000587, A000670, A064898, A331797.

g:= proc(n) option remember;
     `if`(n=0, 0, n*(g(n-1)-(-1)^n))
    end:
b:= proc(n, m) option remember; `if`(n=0,
      g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 23 2023

nmax = 20; CoefficientList[Series[(Exp[x] - 1) Exp[1 - Exp[x]]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
A000240[n_] := n! Sum[(-1)^k/k!, {k, 0, n - 1}]; a[n_] := Sum[StirlingS2[n, k] A000240[k], {k, 0, n}]; Table[a[n], {n, 0, 20}]
Table[(1/2) Sum[Binomial[n, k] HurwitzLerchPhi[1/2, -k, 0] BellB[n - k, -1], {k, 1, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} Stirling2(n,k) * A000240(k).
a(n) = Sum_{k=1..n} binomial(n,k) * A000670(k) * A000587(n-k).
a(n) ~ n! * exp(-1) / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Jan 26 2020

A335111 a(n) = n! * Sum_{k=0..n-1} (-2)^k / k!.

Original entry on oeis.org

0, 1, -2, 6, -8, 40, 48, 784, 5248, 49536, 490240, 5403904, 64822272, 842742784, 11798284288, 176974510080, 2831591636992, 48137058942976, 866467058614272, 16462874118651904, 329257482362552320, 6914407129635618816, 152116956851937476608, 3498690007594658430976

Offset: 0

Ilya Gutkovskiy, May 23 2020

sign

Inverse binomial transform of A000240.

Cf. A000023, A000240, A008290, A066534, A087981, A122803.

Table[n! Sum[(-2)^k/k!, {k, 0, n - 1}], {n, 0, 23}]
nmax = 23; CoefficientList[Series[Sum[k! x^k/(1 + 2 x)^(k + 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[x Exp[-2 x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = n! * sum(k=0, n-1, (-2)^k / k!); \\ Michel Marcus, May 23 2020

G.f.: Sum_{k>=1} k! * x^k / (1 + 2*x)^(k + 1).
E.g.f.: x*exp(-2*x) / (1 - x).
a(n) = A000023(n) - A122803(n).
a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Jun 08 2022
a(n) = Sum_{k=0..n} (-1)^k * k * A008290(n,k). - Alois P. Heinz, May 20 2023

A348590 Number of endofunctions on [n] with exactly one isolated fixed point.

Original entry on oeis.org

0, 1, 0, 9, 68, 845, 12474, 218827, 4435864, 102030777, 2625176150, 74701061831, 2329237613988, 78972674630005, 2892636060014050, 113828236497224355, 4789121681108775344, 214528601554419809777, 10193616586275094959534, 512100888749268955942015

Offset: 0

Alois P. Heinz, Dec 20 2021

nonn

			a(3) = 9: 122, 133, 132, 121, 323, 321, 113, 223, 213.

Alois P. Heinz, Table of n, a(n) for n = 0..386

Column k=1 of A350212.

Cf. A000035, A000240, A001865, A250105.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, t) option remember; `if`(n=0, t, add(g(i)*
      b(n-i, `if`(i=1, 1, t))*binomial(n-1, i-1), i=1+t..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}] ;
b[n_, t_] := b[n, t] = If[n == 0, t, Sum[g[i]*
     b[n - i, If[i == 1, 1, t]]*Binomial[n - 1, i - 1], {i, 1 + t, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)

a(n) mod 2 = A000035(n).

A156788 Triangle T(n, k) = binomial(n, k)A000166(n-k)k^n with T(0, 0) = 1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 4, 0, 3, 0, 27, 0, 8, 96, 0, 256, 0, 45, 640, 2430, 0, 3125, 0, 264, 8640, 29160, 61440, 0, 46656, 0, 1855, 118272, 688905, 1146880, 1640625, 0, 823543, 0, 14832, 1899520, 16166304, 41287680, 43750000, 47029248, 0, 16777216, 0, 133497, 34172928, 438143580, 1453326336, 2214843750, 1693052928, 1452729852, 0, 387420489

Offset: 0

Roger L. Bagula, Feb 15 2009

			Triangle begins as:
  1;
  0,     1;
  0,     0,       4;
  0,     3,       0,       27;
  0,     8,      96,        0,      256;
  0,    45,     640,     2430,        0,     3125;
  0,   264,    8640,    29160,    61440,        0,    46656;
  0,  1855,  118272,   688905,  1146880,  1640625,        0, 823543;
  0, 14832, 1899520, 16166304, 41287680, 43750000, 47029248,      0, 16777216;

J. Riordan, Combinatorial Identities, Wiley, 1968, p.194.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000166, A000240, A000312, A137341.

A000166[n_]:= A000166[n]= If[n==0, 1, n*A000166[n-1] + (-1)^n];
T[n_, k_]:= If[n==0, 1, Binomial[n, k]*A000166[n-k]*k^n];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 10 2021 *)

def A000166(n): return 1 if (n==0) else n*A000166(n-1) + (-1)^n
def A156788(n,k): return 1 if (n==0) else binomial(n,k)*k^n*A000166(n-k)
flatten([[A156788(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 10 2021

T(n, k) = binomial(n, k)*A000166(n-k)*k^n with T(0, 0) = 1.
T(n, k) = binomial(n, k)*b(n-k)*k^n, where b(n) = n*b(n-1) + (-1)^n and b(0) = 1.
Sum_{k=0..n} T(n, k) = A137341(n).
From G. C. Greubel, Jun 10 2021: (Start)
T(n, 1) = A000240(n).
T(n, n) = A000312(n). (End)

Edited by G. C. Greubel, Jun 10 2021

A161129 Triangle read by rows: T(n,k) is the number of non-derangements of {1,2,...,n} for which the difference between the largest and smallest fixed points is k (n>=1; 0 <= k <= n-1).

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 8, 3, 2, 2, 45, 8, 9, 8, 6, 264, 45, 44, 42, 36, 24, 1855, 264, 265, 256, 234, 192, 120, 14832, 1855, 1854, 1810, 1704, 1512, 1200, 720, 133497, 14832, 14833, 14568, 13950, 12864, 11160, 8640, 5040, 1334960, 133497, 133496, 131642, 127404

Offset: 1

Emeric Deutsch, Jul 18 2009

Row sums give the number of non-derangement permutations of {1,2,...,n} (A002467).
T(n,0) = A000240(n) = number of permutations of {1,2,...,n} with exactly 1 fixed point.
T(n,1) = A000240(n-1).
T(n,2) = A000166(n-1) (the derangement numbers).
T(n,3) = A018934(n-1).
Sum_{k=0..n-1} k*T(n,k) = A161130(n).

			T(4,1)=3 because we have 1243, 4231, and 2134; T(4,2)=2 because we have 1432 and 3214; T(5,4)=6 because we have 1xyz5 where xyz is any permutation of 234.
Triangle starts:
    1;
    0,  1;
    3,  0,  1;
    8,  3,  0,  1;
   45,  8,  9,  8,  6;
  264, 45, 44, 42, 36, 24;

E. Deutsch and S. Elizalde, The largest and the smallest fixed points of permutations, arXiv:0904.2792 [math.CO], 2009.

Cf. A000166, A000240, A002467, A018934, A161130.

d[0] := 1: for n to 15 do d[n] := n*d[n-1]+(-1)^n end do: T := proc (n, k) if k = 0 then n*d[n-1] elif k < n then (n-k)*(sum(binomial(k-1, j)*d[n-2-j], j = 0 .. k-1)) else 0 end if end proc: for n to 10 do seq(T(n, k), k = 0 .. n-1) end do; # yields sequence in triangular form

d = Subfactorial;
T[n_, 0] := n*d[n - 1];
T[n_, k_] := (n - k)*Sum[d[n - j - 2]*Binomial[k - 1, j], {j, 0, k - 1}];
Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Nov 28 2017 *)

T(n,0) = n*d(n-1); T(n,k) = (n-k)*Sum_{j=0..k-1}d(n-2-j)*binomial(k-1,j) for 1 <= k <= n-1, where d(i)=A000166(i) are the derangement numbers.

cp's OEIS Frontend

A161130 Sum of the differences between the largest and the smallest fixed points over all non-derangement permutations of {1,2,...,n}.

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A182390 a(0)=0, a(n) = (a(n-1) * n) XOR n.

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A211229 Matrix inverse of lower triangular array A211226.

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A235378 a(n) = (-1)^n*(n! - (-1)^n).

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A318365 Expansion of e.g.f. exp(x*exp(-x)/(1 - x)).

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A331796 E.g.f.: (exp(x) - 1) * exp(1 - exp(x)) / (2 - exp(x)).

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A335111 a(n) = n! * Sum_{k=0..n-1} (-2)^k / k!.

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A348590 Number of endofunctions on [n] with exactly one isolated fixed point.

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A156788 Triangle T(n, k) = binomial(n, k)A000166(n-k)k^n with T(0, 0) = 1, read by rows.