A055596 -id:A055596 - cp's OEIS frontend

A092582 Triangle read by rows: T(n,k) is the number of permutations p of [n] having length of first run equal to k.

1, 1, 1, 3, 2, 1, 12, 8, 3, 1, 60, 40, 15, 4, 1, 360, 240, 90, 24, 5, 1, 2520, 1680, 630, 168, 35, 6, 1, 20160, 13440, 5040, 1344, 280, 48, 7, 1, 181440, 120960, 45360, 12096, 2520, 432, 63, 8, 1, 1814400, 1209600, 453600, 120960, 25200, 4320, 630, 80, 9, 1

Offset: 1

Emeric Deutsch and Warren P. Johnson (wjohnson(AT)bates.edu), Apr 10 2004

Row sums are the factorial numbers (A000142). First column is A001710.
T(n,k) = number of permutations of [n] in which 1,2,...,k is a subsequence but 1,2,...,k,k+1 is not. Example: T(4,2)=8 because 1324, 1342, 1432, 4132, 3124, 3142, 3412 and 4312, are the only permutations of [4] in which 12 is a subsequence but 123 is not. - Emeric Deutsch, Nov 12 2004
T(n,k) is the number of deco polyominoes of height n with k cells in the last column. (A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column). - Emeric Deutsch, Jan 06 2005
T(n,k) is the number of permutations p of [n] for which the smallest i such that p(i)Emeric Deutsch, Feb 23 2008
Adding columns 2,4,6,... one obtains the derangement numbers 0,1,2,9,44,... (A000166). See the Bona reference (p. 118, Exercises 41,42). - Emeric Deutsch, Feb 23 2008
Matrix inverse of A128227*A154990. - Mats Granvik, Feb 08 2009
Differences in the columns of A173333 which counts the n-permutations with an initial ascending run of length at least k. - Geoffrey Critzer, Jun 18 2017
The triangle with each row reversed is A130477. - Michael Somos, Jun 25 2017

			T(4,3) = 3 because 1243, 1342 and 2341 are the only permutations of [4] having length of first run equal to 3.
     1;
     1,    1;
     3,    2,   1;
    12,    8,   3,   1;
    60,   40,  15,   4,  1;
   360,  240,  90,  24,  5,  1;
  2520, 1680, 630, 168, 35,  6,  1;
  ...

M. Bona, Combinatorics of Permutations, Chapman&Hall/CRC, Boca Raton, Florida, 2004.

Alois P. Heinz, Rows n = 1..141, flattened
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Colin Defant and James Propp, Quantifying Noninvertibility in Discrete Dynamical Systems, arXiv:2002.07144 [math.CO], 2020.
Emeric Deutsch and W. P. Johnson, Create your own permutation statistics, Math. Mag., 77, 130-134, 2004.

Cf. A002260. - Gary W. Adamson, Feb 22 2012
Cf. A000166, A002627, A055596, A092580, A092956, A130477.
Cf. A000027, A001710, A001715, A001720, A001725, A005563.
Cf. A000522.

Flat(List([1..11],n->Concatenation([1],List([1..n-1],k->Factorial(n)*k/Factorial(k+1))))); # Muniru A Asiru, Jun 10 2018

A092582:= func< n,k | k eq n select 1 else k*Factorial(n)/Factorial(k+1) >;
[A092582(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Sep 06 2022

Drop[Drop[Abs[Map[Select[#, # < 0 &] &, Map[Differences, nn = 10; Range[0, nn]! CoefficientList[Series[(Exp[y x] - 1)/(1 - x), {x, 0, nn}], {x, y}]]]], 1], -1] // Grid (* Geoffrey Critzer, Jun 18 2017 *)

{T(n, k) = if( n<1 || k>n, 0, k==n, 1, n! * k /(k+1)!)}; /* Michael Somos, Jun 25 2017 */

def A092582(n,k): return 1 if (k==n) else k*factorial(n)/factorial(k+1)
flatten([[A092582(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Sep 06 2022

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

Inverse of:

-1, 1;

-1, -2, 1;

-1, -2, -3, 1;

-1, -2, -3, -4, 1;

... where A002260 = (1; 1,2; 1,2,3; ...). - Gary W. Adamson, Feb 22 2012

T(2n,n) = A092956(n-1) for n>0. - Alois P. Heinz, Jun 19 2017

From Alois P. Heinz, Dec 17 2021: (Start)

Sum_{k=1..n} k * T(n,k) = A002627(n).

|Sum_{k=1..n} (-1)^k * T(n,k)| = A055596(n) for n>=1. (End)

From G. C. Greubel, Sep 06 2022: (Start)

T(n, 1) = A001710(n).

T(n, 2) = 2*A001715(n) + [n=2]/3, n >= 2.

T(n, 3) = 3*A001720(n) + [n=3]/4, n >= 3.

T(n, 4) = 4*A001725(n) + [n=4]/5, n >= 4.

T(n, n-1) = A000027(n-1).

T(n, n-2) = A005563(n-1), n >= 3. (End)

Sum_{k=0..n} (k+1) * T(n,k) = A000522(n). - Alois P. Heinz, Apr 28 2023

A096654 Denominators of self-convergents to 1/(e-2).

Original entry on oeis.org

1, 2, 8, 38, 222, 1522, 11986, 106542, 1054766, 11506538, 137119578, 1772006854, 24681524038, 368577425634, 5874202721042, 99515904921182, 1785757627196766, 33835407673201882, 675016383080377546, 14143200407398386678, 310507536216973671158, 7128173005328786885714

Offset: 0

Clark Kimberling, Jul 01 2004

The self-continued fraction of r>0 is here introduced as the sequence (b(0), b(1), b(2), ...) defined as follows: put r(0)=r, b(0)=[r(0)] and for n>=1, put r(n)=b(n-1)/(r(n-1)-b(n-1)) and b(n)=[r(n)]. This differs from simple continued fraction, for which r(n)=1/(r(n-1)-b(n-1)). Now r=lim(p(n)/q(n)), where p(0)=b(1), q(0)=1, p(1)=b(0)(b(1)+1), q(1)=b(1) and for n>=2, p(n)=b(n)*p(n-1)+b(n-1)*p(n-2), q(n)=b(n)*q(n-1)+b(n-1)*q(n-2); p(0),p(1),... are the numerators of the self-convergents to r; q(0),q(1),... are the denominators of the self-convergents to r. Thus A096654 is given by a(n)=(n+1)*a(n-1)+n*a(n-2), a(0)=1, a(1)=2.

Number of increasing runs of odd length in all permutations of [n+1]. Example: a(2) = 8 because we have (123), 13(2), (3)12, (2)13, 23(1), (3)(2)(1) (the runs of odd length are shown between parentheses). - Emeric Deutsch, Aug 29 2004

			a(2)=q(2)=3*2+2*1=8, a(3)=q(3)=4*8+3*2=38. The convergents p(0)/q(0) to p(4)/q(4) are 1/1, 3/2, 11/8, 53/38, 309/222.

Cf. A000255, A096655, A096656, A096657, A096658, A097591, A233590.

G:=(3-x-2*(1+x)*exp(-x))/(1-x)^3: Gser:=series(G,x=0,22): 1,seq(n!*coeff(Gser,x^n),n=1..21);

With[{g = (3 - x - 2*(1 + x)*Exp[-x])/(1 - x)^3},CoefficientList[Series[g, {x, 0, 21}], x]*Table[k!, {k, 0, 21}]] (* Shenghui Yang, Oct 15 2024 *)

x='x+O('x^66); Vec(serlaplace((3-x-2*(1+x)*exp(-x))/(1-x)^3)) /* Joerg Arndt, Aug 06 2012 */

prpr = 1
prev = 2
for n in range(2, 77):
    print(prpr, end=', ')
    curr = (n+1)*prev + n*prpr
    prpr = prev
    prev = curr
# Alex Ratushnyak, Aug 05 2012

a(n) = (n+1)*a(n-1) + n*a(n-2), with a(0)=1, a(1)=2. - Alex Ratushnyak, Aug 05 2012
E.g.f.: (3-x-2*(1+x)*exp(-x))/(1-x)^3. - Emeric Deutsch, Aug 29 2004
From Gary Detlefs, Apr 12 2010: (Start)
a(n) = A055596(n+1) + A055596(n+2).
a(n) = (n+1)!+(n+2)! -2*( A000166(n+1) + A000166(n+2)).
a(n) = (n+1)! - 2*floor(((n+1)!+1)/e) + (n+2)!-2*floor(((n+2)!+1)/e). (End)
a(n) = ((n+3)!-2*floor(((n+3)!+1)/e))/(n+2). - Gary Detlefs, Jul 11 2010 [corrected by Gary Detlefs, Oct 26 2020]
a(n) = Sum_{k=1..n+1} A097591(n+1,k). - Alois P. Heinz, Jul 03 2019

More terms from Emeric Deutsch, Aug 29 2004

A227918 Sum over all permutations beginning and ending with ascents, and without double ascents on n elements and each permutation contributes 2 to the power of the number of double descents.

Original entry on oeis.org

1, 0, 5, 22, 137, 956, 7653, 68874, 688745, 7576192, 90914309, 1181886014, 16546404201, 248196063012, 3971137008197, 67509329139346, 1215167924508233, 23088190565656424, 461763811313128485, 9697040037575698182, 213334880826665360009, 4906702259013303280204, 117760854216319278724901

Offset: 2

Richard Ehrenborg, Oct 08 2013

nonn

			a(4) = 5 since the sum is over the five permutations 1324, 1423, 2314, 2413 and 3412, and each of them contribute 1 to the sum, since none of them has a double descent.

R. Ehrenborg and J. Jung, Descent pattern avoidance, Adv. in Appl. Math., 49 (2012) 375-390.

Cf. A000166, A230071, A055596.

a[2] = 1; a[n_] := n*a[n - 1] + 1 + 4 (-1)^n;  Table[a[n], {n, 2, 20}] (* Wesley Ivan Hurt, May 04 2014 *)

E.g.f.: (exp(x) - 4 + 4*exp(-x))/(1-x) - 1 + 2*x.
Closest integer to (e - 4 + 4/e)*n! for n > 7.
a(n) = n*a(n-1) + 1 + 4*(-1)^n.
Conjecture: a(n) -n*a(n-1) -a(n-2) +(n-2)*a(n-3) = 0. - R. J. Mathar, Jul 17 2014

A230071 Sum over all permutations without double ascents on n elements and each permutation contributes 2 raised to the power of the number of double descents.

Original entry on oeis.org

0, 0, 2, 6, 26, 130, 782, 5474, 43794, 394146, 3941462, 43356082, 520272986, 6763548818, 94689683454, 1420345251810, 22725524028962, 386333908492354, 6954010352862374, 132126196704385106, 2642523934087702122, 55493002615841744562, 1220846057548518380366

Offset: 0

Richard Ehrenborg, Oct 08 2013

nonn

			For n=3 the a(3)= 6 since the 4 permutations 132, 213, 231, 312 all contribute 1 and 321 contributes 2 to the sum. Note when n=4, the permutation 4321 contributes 4 since it has two double descents.
G.f. = 2*x^2 + 6*x^3 + 26*x^4 + 130*x^5 + 782*x^6 + 5474*x^7 + 43794*x^8 + ...

R. Ehrenborg and J. Jung, Descent pattern avoidance, Adv. in Appl. Math., 49 (2012) 375-390.

Cf. A000166, A002627, A080227, A055596, A227918.

a := proc(n) if n < 2 then 0 elif n = 2 then 2 else (2-n)*a(n-3)+a(n-2)+n*a(n-1) fi end: seq(a(n), n=0..9); # Peter Luschny, May 30 2014

a[0] = 0; a[n_] := a[n] = n a[n-1] + (-1)^n + 1;
Array[a, 23, 0] (* Jean-François Alcover, Jul 08 2019, after A080227 *)

E.g.f.: (exp(x)+exp(-x)-2)/(1-x).
a(n) = closest integer to (e-2+1/e)*n! for n > 3.
a(n) = (2-n)*a(n-3) + a(n-2) + n*a(n-1) for n > 2.
a(n) = 2*A080227(n).
a(n) = sum(0<=kA002627(k)). - Peter Luschny, May 30 2014
0 = a(n)*(+a(n+1) - a(n+2) - 3*a(n+3) + a(n+4)) + a(n+1)*(+a(n+1) + a(n+2) - 2*a(n+3)) + a(n+2)*(+a(n+2) + a(n+3) - a(n+4)) + a(n+3)*(+a(n+3)) if n>=0. - Michael Somos, May 30 2014

a(0) and a(1) prepended, partially edited. - Peter Luschny, May 30 2014

A355229 E.g.f. A(x) satisfies A'(x) = 1 - log(1-x) * A(x).

Original entry on oeis.org

0, 1, 0, 2, 3, 16, 65, 365, 2261, 16240, 131097, 1182013, 11779537, 128737088, 1532051287, 19731964705, 273556185109, 4062828620256, 64368863326717, 1083795820014261, 19327395713028985, 363940825109825200, 7216468161637890899, 150304143164083288441

Offset: 0

Seiichi Manyama, Jun 25 2022

nonn

Cf. A055596, A087650, A355230, A355231.

nmax = 25; CoefficientList[Series[(1-x)^(1-x) / E^(1-x) * Integrate[E^(1-x) / (1-x)^(1-x), x], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 25 2022 *)

a_vector(n) = my(v=vector(n)); v[1]=1; for(i=1, n-1, v[i+1]=sum(j=1, i-1, (j-1)!*binomial(i, j)*v[i-j])); concat(0, v);

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

E.g.f.: (1-x)^(1-x) / exp(1-x) * Integral(exp(1-x) / (1-x)^(1-x) dx). - Vaclav Kotesovec, Jun 25 2022

A216441 a(n) = n! mod !n.

Original entry on oeis.org

0, 0, 6, 32, 190, 1332, 10654, 95888, 958878, 10547660, 126571918, 1645434936, 23036089102, 345541336532, 5528661384510, 93987243536672, 1691770383660094, 32143637289541788, 642872745790835758, 13500327661607550920, 297007208555366120238, 6831165796773420765476

Offset: 2

Michel Lagneau, Sep 07 2012

nonn

!n is a subfactorial number (A000166).

			a(5) = 5! mod !5 = 120 mod 44 = 32. - _Indranil Ghosh_, Feb 15 2017

Indranil Ghosh, Table of n, a(n) for n = 2..449

Cf. A000166, A000142, A055596.

with(numtheory): f:=n->sum(n!*(((-1)^k)*1/k!), k=0..n):for n from 1 to 30 do:  x:=irem(n!,f(n)): printf(`%d, `, x):od:

Table[Mod[n !, Subfactorial[n]],{n,100} ]

A179539 a(0) = 1, a(1) = 0, a(n) = 2n(a(n-1) + a(n-2)), n > 1.

Original entry on oeis.org

1, 0, 4, 24, 224, 2480, 32448, 488992, 8343040, 158976576, 3346392320, 77118115712, 1931148192768, 52214924020480, 1516090021970944, 47049148379742720, 1554087628854837248, 54438650425975718912, 2015738569973900021760, 78666734375195278145536, 3227298917806767126691840

Offset: 0

Gary Detlefs, Jul 18 2010

			a(2)= 4*(0+1)=4, a(3)=6*(4+0)=24, a(4)=8*(24+4)=224...

Cf. A055596.

D-finite with recurrence: 2*n*a(n-2) + 2*n*a(n - 1) - a(n) = 0. Georg Fischer, Mar 31 2025

Definition corrected by Bruno Berselli, Jul 20 2010

Typo in a(7) corrected by Georg Fischer, Mar 31 2025

A361938 a(0)=1, a(1)=0; a(n) = floor(n/2)*(a(n-1) + a(n-2)).

Original entry on oeis.org

1, 0, 1, 1, 4, 10, 42, 156, 792, 3792, 22920, 133560, 938880, 6434640, 51614640, 406344960, 3663676800, 32560174080, 326014657920, 3227173488000, 35531881459200, 387590549472000, 4654346740243200, 55461310186867200, 721387883125324800, 9322190319746304000

Offset: 0

Davide Oliveri, Mar 31 2023

nonn

For n <= 1000000, n prime divides a(n) only when n=5 and n composite does not divide a(n) only when n = 9. Is this always so?

			a(0) = 1;
a(1) = 0;
a(2) = floor(2/2)*(a(1) + a(0)) = 1;
a(3) = floor(3/2)*(a(2) + a(1)) = 1;
a(4) = floor(4/2)*(a(3) + a(2)) = 4;
a(5) = floor(5/2)*(a(4) + a(3)) = 10.

Cf. A055596.

a[0] = 1; a[1] = 0; a[n_] := a[n] = Floor[n/2] * (a[n - 1] + a[n - 2]); Array[a, 30, 0] (* Amiram Eldar, Apr 05 2023 *)

def seqx_it(n):
  a0 = 1
  a1 = 0
  sequence_store = [a0,a1]
  for i in range (2,n):
    a2 = (i//2) * (a1 + a0)
    sequence_store.append(a2)
    a0 = a1
    a1 = a2
  return sequence_store

a(0)=1, a(1)=0; a(n) = floor(n/2)*(a(n-1) + a(n-2)).

Showing 1-8 of 8 results.

cp's OEIS Frontend

A092582 Triangle read by rows: T(n,k) is the number of permutations p of [n] having length of first run equal to k.

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A096654 Denominators of self-convergents to 1/(e-2).

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A227918 Sum over all permutations beginning and ending with ascents, and without double ascents on n elements and each permutation contributes 2 to the power of the number of double descents.

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A230071 Sum over all permutations without double ascents on n elements and each permutation contributes 2 raised to the power of the number of double descents.

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A355229 E.g.f. A(x) satisfies A'(x) = 1 - log(1-x) * A(x).

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A216441 a(n) = n! mod !n.

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A179539 a(0) = 1, a(1) = 0, a(n) = 2*n*(a(n-1) + a(n-2)), n > 1.

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A179539 a(0) = 1, a(1) = 0, a(n) = 2n(a(n-1) + a(n-2)), n > 1.