A013999 -id:A013999 - cp's OEIS frontend

A227037 Partial sums of A013999.

1, 2, 4, 12, 54, 312, 2136, 16800, 149160, 1475280, 16081920, 191530080, 2473999920, 34446303360, 514240110720, 8193624284160, 138780284791680, 2489891543596800, 47169750454848000, 940914453958617600, 19712190644360121600

Offset: 0

Emanuele Munarini, Jul 01 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A013999.

a:= proc(n) option remember; `if`(n<3, 2^n,
      (n+3)*a(n-1) -2*(n+1)*a(n-2) +(n+1)*a(n-3))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 01 2013

Table[Sum[Sum[Binomial[j-k+1,k]*(-1)^k*(j-k+1)!,{k,0,Floor[(j+1)/2]}],{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 06 2013 *)

makelist(sum(sum(binomial(j-k+1,k)*(-1)^k*(j-k+1)!, k,0,floor((j+1)/2)), j, 0, n), n, 0, 20);

a(n) = sum(A013999(k), k=0..n).
a(n) = sum(sum(C(j-k+1,k)*(-1)^k*(j-k+1)!, k=0..floor((j+1)/2)), j=0..n).
Recurrence: a(n+4) -(n+8)*a(n+3) +(3*n+16)*a(n+2) -(3*n+13)*a(n+1) +(n+4)*a(n) = 0.
G.f.: Sum_{k>=0} (k+1)!*(x-x^2)^k.
a(n) = (n+3)*a(n-1)-2*(n+1)*a(n-2)+(n+1)*a(n-3) for n>2, a(n) = 2^n for n<=2. - Alois P. Heinz, Jul 01 2013
a(n) ~ n!*n/exp(1). - Vaclav Kotesovec, Jul 06 2013

A227096 Self-convolution of A013999.

Original entry on oeis.org

1, 2, 5, 20, 104, 632, 4396, 34680, 307236, 3026472, 32849364, 389704800, 5017492320, 69678231552, 1038078389376, 16513758904320, 279354776803200, 5007072973075200, 94783054774919040, 1889504358498754560, 39565281716813111040, 868194780280625779200

Offset: 0

Emanuele Munarini, Jul 01 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A013999.

a:= proc(n) option remember; `if`(n<6, [1, 2, 5, 20, 104, 632][n+1],
      ((3*n+10)*(n+3)*a(n-1) -(n+13)*(n+2)^2*a(n-2)
       +(n+3)*(4*n^2+19*n+2)*a(n-3) -2*(n+2)*(3*n^2+6*n-4)*a(n-4)
       +(4*n^3+8*n^2-12*n-4)*a(n-5) -n*(n+3)*(n-2)*a(n-6))/(2*n+4))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 01 2013

f[n_] := Sum[Binomial[n-k+1, k] (-1)^k (n-k+1)!, {k, 0, Quotient[n+1, 2]}];
a[n_] := Sum[f[k] f[n-k], {k, 0, n}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 14 2023 *)

f(n):=sum(binomial(n-k+1, k)*(-1)^k*(n-k+1)!, k, 0, floor((n+1)/2)); a(n):=sum(f(k)*f(n-k), k, 0, n); makelist(a(n), n, 0, 20);

a(n) = sum(A013999(k)*A013999(n-k), k=0..n).
G.f.: sum(B(k)*k!*x^(k-2)*(1-x)^k, k>=2), where B(k) = sum(1/C(k,i), i=1..k-1).
a(n) ~ 2*n*n!/exp(1). - Vaclav Kotesovec, Jul 08 2013

A227094 Binomial transform of A013999.

Original entry on oeis.org

1, 2, 5, 18, 91, 574, 4199, 34650, 318645, 3237034, 36041657, 436713506, 5722676895, 80654047942, 1216703923147, 19562850695690, 333991034593833, 6034449711055890, 115036771019660269, 2307582082535387570, 48588759062255598563, 1071533741191907032590

Offset: 0

Emanuele Munarini, Jul 01 2013

nonn

Cf. A013999.

a:= proc(n) option remember; `if`(n<4, [1, 2, 5, 18][n+1],
      (n+6)*a(n-1)-(5*n+7)*a(n-2)+(8*n-7)*a(n-3)-(4*n-12)*a(n-4))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 01 2013

a[n_] := a[n] = If[n < 4, {1, 2, 5, 18}[[n + 1]], (n + 6)*a[n - 1] - (5*n + 7)*a[n - 2] + (8*n - 7)*a[n - 3] - (4*n - 12)*a[n - 4]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 04 2018, from Maple *)

f(n):=sum(binomial(n-k+1,k)*(-1)^k*(n-k+1)!, k, 0, floor((n+1)/2)); a(n):=sum(binomial(n,k)*f(k), k, 0, n); makelist(a(n), n,0,20);

a(n) = sum(C(n,k)*A013999(k), k=0..n).
G.f.: sum(k!*x^(k-1)*(1-2*x)^k/(1-x)^(2*k), k>=1).
a(n) ~ n * n!. - Vaclav Kotesovec, Nov 02 2023

A127548 O.g.f.: Sum_{n>=0} n!*(x/(1+x)^2)^n.

Original entry on oeis.org

1, 1, 0, 1, 4, 19, 112, 771, 6088, 54213, 537392, 5867925, 69975308, 904788263, 12607819040, 188341689287, 3002539594128, 50878366664393, 913161208490016, 17304836525709097, 345279674107957524, 7235298537356113339

Offset: 0

Vladeta Jovovic, Jun 27 2007

a(n+1) = inverse binomial transform of A013999 = Sum_{k=0..n} binomial(n,k)*(-1)^(n-k)*A013999(k). - Emanuele Munarini, Jul 01 2013

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A000179, A078480, A013999, A000271, A000904, A078509.

A127548 := proc(n) if n = 0 then 1 ; else add(factorial(s)*(-1)^(n-s)*binomial(s+n-1,2*s-1),s=1..n) ; fi ; end: for n from 0 to 20 do printf("%d,",A127548(n)) ; od ; # R. J. Mathar, Jul 13 2007

nn = 21; CoefficientList[Series[Sum[n!*(x/(1 + x)^2)^n, {n, 0, nn}], {x, 0, nn}], x] (* Michael De Vlieger, Sep 04 2016 *)

import math
def binomial(n,m):
    a=1
    for k in range(n-m+1,n+1):
        a *= k
    return a//math.factorial(m)
def A127548(n):
    if n == 0:
        return 1
    a=0
    for s in range(1,n+1):
        a += (-1)**(n-s)*binomial(s+n-1,2*s-1)*math.factorial(s)
    return a
for n in range(30):
    print(A127548(n))
# R. J. Mathar, Oct 20 2009

a(n) = Sum_{s=1..n} (-1)^(n-s)*s!*C(s+n-1,2s-1) if n>=1, where C(a,b)=binomial(a,b). - R. J. Mathar, Jul 13 2007
G.f.: Q(0) where Q(k) = 1 + (2*k + 1)*x/( (1+x)^2- 2*x*(1+x)^2*(k+1)/(2*x*(k+1) + (1+x)^2/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 08 2013
a(n) = A000271(n) + A000271(n-1). - Peter Bala, Sep 02 2016
a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Oct 31 2017

More terms from R. J. Mathar, Jul 13 2007

More terms from R. J. Mathar, Oct 20 2009

A184185 Number of permutations of {1,2,...,n} having no cycles of the form (i, i+1, i+2, ..., i+j-1) (j >= 1).

Original entry on oeis.org

1, 0, 0, 1, 6, 34, 216, 1566, 12840, 117696, 1193760, 13280520, 160841520, 2107021680, 29689833600, 447821503920, 7199590366080, 122907276334080, 2220524598297600, 42328747652446080, 849064844592518400, 17877531486897734400, 394246607165708774400

Offset: 0

Emeric Deutsch, Feb 16 2011 (based on communication from Vladeta Jovovic)

nonn

a(n) = A184184(n,0).

			a(4)=6 because we have (13)(24), (1432), (1342), (1423), (1243), and (1324).

Seiichi Manyama, Table of n, a(n) for n = 0..450
Patxi Laborde Zubieta, Occupied corners in tree-like tableaux, arXiv preprint arXiv:1505.06098 [math.CO], 2015.

Cf. A013999, A184184.

a := proc(n) add((-1)^(n-i)*factorial(i)*binomial(i+1, n-i), i = ceil((1/2)*n-1/2) .. n) end proc: seq(a(n), n = 0 .. 22);

a[n_] := Sum[(-1)^(n-i)*i!*Binomial[i+1, n-i], {i, Ceiling[(n-1)/2], n}];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Nov 29 2017, from Maple *)

a(n) = sum(k=n\2, n, (-1)^(n-k)*k!*binomial(k+1, n-k)); \\ Seiichi Manyama, Nov 30 2021

a(n) = if(n<3, 0^n, (n+2)*a(n-1)-2*(n-1)*a(n-2)+(n-2)*a(n-3)); \\ Seiichi Manyama, Nov 30 2021

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, k!*x^k*(1-x)^(k+1))) \\ Seiichi Manyama, Nov 30 2021

G.f.: (1-z)*F(z-z^2), where F(z) = Sum_{j>=0} j!*z^j (private communication from Vladeta Jovovic, May 26 2009).
a(n) = Sum_{i=ceiling((n-1)/2)..n} (-1)^(n-i)*i!*binomial(i+1,n-i).
G.f.: 1/Q(0), where Q(k) = 1 + x/(1-x) - x*(k+1)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
a(n) ~ n! / exp(1) * (1 - 1/n - 1/(2*n^2) - 2/(3*n^3) - 23/(24*n^4) - 151/(120*n^5) - 119/(720*n^6) + 14789/(1260*n^7) + 1223843/(13440*n^8) + ...). - Vaclav Kotesovec, Nov 30 2021
a(n) = (n+2) * a(n-1) - 2 * (n-1) * a(n-2) + (n-2) * a(n-3) for n > 2. - Seiichi Manyama, Nov 30 2021

A307495 Expansion of Sum_{k>=0} k!((1 - sqrt(1 - 4x))/2)^k.

Original entry on oeis.org

1, 1, 3, 12, 57, 312, 1950, 13848, 111069, 998064, 9957186, 109305240, 1309637274, 17006109072, 237888664572, 3566114897520, 57030565449765, 969154436550240, 17439499379433690, 331268545604793240, 6624013560942038670, 139080391965533653200, 3059323407592802838180, 70355685298375014175440

Offset: 0

Ilya Gutkovskiy, Apr 10 2019

nonn

Catalan transform of A000142 (factorial numbers).
From Peter Bala, Jan 27 2020: (Start)
This sequence is the main diagonal of the lower triangular array formed by putting the sequence of factorial numbers in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.
    1
    1    1
    2    3    3
    6    9   12   12
   24   33   45   57   57
  120  153  198  255  312  312
  ...
Alternatively, the sequence can be obtained by multiplying the sequence of factorial numbers by the array A106566.
(End)

P. Bala, A note on the Catalan transform of a sequence

Cf. A000108, A000142, A013999, A100100, A106566, A307496.

nmax = 23; CoefficientList[Series[Sum[k! ((1 - Sqrt[1 - 4 x])/2)^k, {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[1/(1 + ContinuedFractionK[-Floor[(k + 1)/2] (1 - Sqrt[1 - 4 x])/2, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
Join[{1}, Table[1/n Sum[Binomial[2n - k - 1, n - k] k k!, {k, n}], {n, 23}]]

G.f.: 1 /(1 - x*c(x)/(1 - x*c(x)/(1 - 2*x*c(x)/(1 - 2*x*c(x)/(1 - 3*x*c(x)/(1 - 3*x*c(x)/(1 - ...))))))), a continued fraction, where c(x) = g.f. of Catalan numbers (A000108).
Sum_{n>=0} a(n)*(x*(1 - x))^n = g.f. of A000142.
a(n) = (1/n) * Sum_{k=1..n} binomial(2*n-k-1,n-k)*k*k! for n > 0.
a(n) ~ exp(1) * n!. - Vaclav Kotesovec, Aug 10 2019

A370508 Expansion of Sum_{k>=0} k! * ( x * (1-x^3) )^k.

Original entry on oeis.org

1, 1, 2, 6, 23, 116, 702, 4944, 39722, 358578, 3593664, 39595440, 475746474, 6190838544, 86740334160, 1301939398080, 20842001737224, 354469125185880, 6382790173842480, 121310821042966800, 2426863248540057480, 50975836645480342560, 1121691979824460425360

Offset: 0

Seiichi Manyama, Feb 20 2024

Cf. A013999, A212580.

Cf. A370510.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x*(1-x^3))^k))

a(n) = sum(k=0, n\4, (-1)^k*(n-3*k)!*binomial(n-3*k, k));

a(n) = Sum_{k=0..floor(n/4)} (-1)^k * (n-3*k)! * binomial(n-3*k,k).

A184184 Triangle read by rows: T(n,k) is the number of permutations of [n] having k adjacent cycles (0 <= k <= n). An adjacent cycle is a cycle of the form (i, i+1, i+2, ...) (including 1-element cycles).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 6, 8, 6, 3, 1, 34, 42, 27, 12, 4, 1, 216, 258, 156, 64, 20, 5, 1, 1566, 1824, 1068, 420, 125, 30, 6, 1, 12840, 14664, 8400, 3220, 930, 216, 42, 7, 1, 117696, 132360, 74580, 28080, 7950, 1806, 343, 56, 8, 1, 1193760, 1326120, 737640, 273960, 76440, 17094, 3192, 512, 72, 9, 1

Offset: 0

Emeric Deutsch, Feb 16 2011 (based on communication from Vladeta Jovovic)

Sum of entries in row n is n!.
T(n,0) = A184185(n).
T(n,1) = A013999(n-1).
Sum_{k>=0} k*T(n,k) = 1! + 2! + ... + n! = A007489(n).

			T(3,2) = 2 because we have (1)(23) and (12)(3).
T(4,2) = 6 because we have (1)(234), (1)(24)(3), (12)(34), (123)(4), (14)(2)(3), and (13)(2)(4).
Triangle starts:
   1;
   0,  1;
   0,  1,  1;
   1,  2,  2,  1;
   6,  8,  6,  3,  1;
  34, 42, 27, 12,  4,  1;

FindStat - Combinatorial Statistic Finder, The number of adjacent cycles of a permutation

Cf. A007489, A013999, A184185.

T := proc (n, k) options operator, arrow: add((-1)^(n-k-i)*factorial(k+i)*binomial(i+1, n-k-i), i = ceil((1/2)*n-(1/2)*k-1/2) .. n-k)/factorial(k) end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form

G.f. of column k is (1/k!)*z^k*(1-z)*Sum_{i>=0} (k+i)!*(z-z^2)^i (private communication from Vladeta Jovovic, May 26 2009).
T(n,k) = (1/k!)*Sum_{i=ceiling((n-k-1)/2)..n-k} (-1)^(n-k-i)*(k+i)!*binomial(i+1, n-k-i).
The bivariate g.f. is G(t,z) = ((1-z)/(1-tz))*F((z-z^2)/(1-tz)), where F(z) = Sum_{j>=0} j!*z^j.

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A227037 Partial sums of A013999.

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A227096 Self-convolution of A013999.

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A227094 Binomial transform of A013999.

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A127548 O.g.f.: Sum_{n>=0} n!*(x/(1+x)^2)^n.

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A184185 Number of permutations of {1,2,...,n} having no cycles of the form (i, i+1, i+2, ..., i+j-1) (j >= 1).

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A307495 Expansion of Sum_{k>=0} k!*((1 - sqrt(1 - 4*x))/2)^k.

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A370508 Expansion of Sum_{k>=0} k! * ( x * (1-x^3) )^k.

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A307495 Expansion of Sum_{k>=0} k!((1 - sqrt(1 - 4x))/2)^k.