Vladeta Jovovic - cp's OEIS frontend

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).

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.

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

A168655 Number of compositions such that the number of parts is divisible by the first part.

Original entry on oeis.org

1, 1, 3, 5, 11, 22, 44, 88, 177, 355, 710, 1419, 2838, 5679, 11363, 22727, 45443, 90862, 181703, 363419, 726903, 1453875, 2907667, 5814880, 11628864, 23256828, 46513965, 93031069, 186068503, 372142797, 744280096, 1488527555, 2976987042, 5953897971, 11907811651

Offset: 1

Vladeta Jovovic, Dec 01 2009

Alois P. Heinz, Table of n, a(n) for n = 1..250

Cf. A079501.

b:= proc(n,t,g) option remember; `if`(n=0,
      `if`(irem(t, g)=0, 1, 0), add(b(n-i, t+1,
      `if`(g=0,i,g)), i=1..n))
    end:
a:= n-> b(n,0,0):
seq(a(n), n=1..40); # Alois P. Heinz, Dec 15 2009

A101510[n_] := Sum[If[Mod[i+1, k+1] == 0, Binomial[n-k, i], 0], {k, 0, n/2}, {i, 0, n-k}]; A168655 =  Join[{1}, Table[A101510[n], {n, 0, 32}] // Differences] (* Jean-François Alcover, Jan 24 2014 *)

G.f.: (1-x)*Sum(x^(2*n-1)/((1-x)^n-x^n),n=1..infinity), First differences of A101510.

a(n) ~ log(2) * 2^(n-1). - Vaclav Kotesovec, May 01 2014

More terms from Alois P. Heinz, Dec 15 2009

A171625 Number of compositions of n such that the number of parts is divisible by the smallest part.

Original entry on oeis.org

1, 1, 3, 7, 15, 29, 58, 118, 242, 493, 997, 2005, 4024, 8071, 16183, 32439, 65003, 130214, 260768, 522084, 1045045, 2091489, 4185209, 8373979, 16753651, 33516419, 67047467, 134118462, 268274858, 536611011, 1073321222, 2146803124, 4293866550, 8588154649

Offset: 1

Vladeta Jovovic, Dec 13 2009

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A168657.

b:= proc(n,t,g) option remember; `if` (n=0, `if` (irem(t, g)=0, 1, 0), add (b(n-i, t+1, min(i, g)), i=1..n)) end: a:= n-> b(n, 0, infinity): seq (a(n), n=1..40);  # Alois P. Heinz, Dec 15 2009

a[n_] := SeriesCoefficient[ Sum[(1-x^k)/(1-x)^k*Sum[x^(k*d), {d, Divisors[k]}], {k, 0, n}], {x, 0, n}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 24 2015 *)

G.f.: Sum_{n>=0}[(1-x^n)/(1-x)^n*Sum_{d|n}x^(n*d)].

a(n) ~ 2^(n-1). - Vaclav Kotesovec, May 01 2014

More terms from Alois P. Heinz, Dec 15 2009

A158615 Expansion of Sum_{n>0} nn!x^n/(1-n!*x^n).

Original entry on oeis.org

1, 5, 19, 105, 601, 4445, 35281, 324897, 3266569, 36360065, 439084801, 5751188913, 80951270401, 1220673888257, 19615124183329, 334777645154817, 6046686277632001, 115243914079782593, 2311256907767808001

Offset: 1

Vladeta Jovovic, Mar 22 2009

a(n) = Sum_{d|n} d*d!^(n/d).

Vaclav Kotesovec, Table of n, a(n) for n = 1..440

Cf. A077365, A265950.

nmax := 40: gf := add( taylor( n*n!*x^n/(1-n!*x^n),x=0,nmax+1),n=1..nmax ) : coeffs(convert(gf,polynom)) ; # R. J. Mathar, Mar 30 2009

nmax=20; Rest[CoefficientList[Series[Sum[k*k!*x^k/(1-k!*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 19 2015 *)

a(n) ~ n * n!. - Vaclav Kotesovec, Dec 19 2015

More terms from R. J. Mathar, Mar 30 2009

A168173 Number of partitions of n in which the sum of reciprocals of parts is less than 1.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 4, 4, 6, 8, 12, 13, 16, 18, 21, 25, 32, 38, 46, 55, 65, 78, 92, 103, 122, 140, 165, 193, 229, 264, 305, 345, 395, 451, 517, 590, 682, 781, 893, 1013, 1165, 1324, 1518, 1717, 1945, 2188, 2468, 2753, 3089, 3457, 3865, 4321, 4856, 5441, 6108, 6831

Offset: 1

Vladeta Jovovic, Nov 19 2009

nonn

Cf. A051908.

a := proc (n) local P, ct, j: with(combinat): P := partition(n): ct := 0: for j to numbpart(n) do if add(1/P[j][i], i = 1 .. nops(P[j])) < 1 then ct := ct+1 else end if end do: ct end proc: seq(a(n), n = 1 .. 60); # Emeric Deutsch, Dec 02 2009

Table[Count[IntegerPartitions[n],?(Total[1/#]<1&)],{n,60}] (* _Harvey P. Dale, Dec 14 2012 *)

Extended by Emeric Deutsch, Dec 02 2009

A159034 Inverse Euler transform of A155200.

Original entry on oeis.org

2, 7, 170, 16380, 6710886, 11453246035, 80421421917330, 2305843009213685760, 268650182136584261045760, 126765060022822940149666965093, 241677817415439249618874010960062650, 1858395433210885261794643189387357732203180, 57560679870263253393868202642364377389525958615670

Offset: 1

Paul D. Hanna, Vladeta Jovovic, Apr 02 2009

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A155200.

Table[Sum[2^(d^2)*MoebiusMu[n/d], {d, Divisors[n]}]/n, {n, 1, 12}] (* Vaclav Kotesovec, Oct 09 2019 *)

a(n)={sumdiv(n, d, 2^(d^2)*moebius(n/d))/n} \\ Andrew Howroyd, Jan 08 2020

a(n) = (1/n)*Sum_{d|n} 2^(d^2)*moebius(n/d).

a(n) ~ 2^(n^2) / n. - Vaclav Kotesovec, Oct 09 2019

Terms a(12) and beyond from Andrew Howroyd, Jan 08 2020

A171628 Number of compositions of n such that the smallest part is divisible by the number of parts.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 3, 4, 6, 8, 11, 15, 19, 22, 25, 30, 37, 47, 62, 83, 108, 136, 168, 205, 247, 295, 354, 429, 524, 642, 789, 972, 1196, 1466, 1789, 2173, 2625, 3155, 3778, 4515, 5391, 6437, 7692, 9201, 11014, 13186, 15780, 18865, 22516, 26818, 31871, 37791

Offset: 1

Vladeta Jovovic, Dec 13 2009

Alois P. Heinz, Table of n, a(n) for n = 1..900

Cf. A168656, A171625.

b:= proc(n,t,g) option remember; `if` (n=0, `if` (irem(g, t)=0, 1, 0), add (b(n-i, t+1, min(i, g)), i=1..n)) end: a:= n-> b(n,0,infinity): seq (a(n), n=1..60); # Alois P. Heinz, Dec 15 2009
A171628 := proc(n) local g,k; g := 0 ; for k from 0 to n do g := g+add (x^(k*d)*(1-x^d)/(1-x)^d,d=numtheory[divisors](k)) ; g := expand(g) ; end do ; coeftayl(g,x=0,n) ; end proc: seq(A171628(n),n=1..60) ; # R. J. Mathar, Dec 14 2009

b[n_, t_, g_] := b[n, t, g] = If[n == 0, If [Mod[g, t] == 0, 1, 0], Sum[b[n - i, t + 1, Min[i, g]], {i, n}]];
a[n_] := b[n, 0, Infinity];
Array[a, 60] (* Jean-François Alcover, May 23 2020, after Alois P. Heinz *)

G.f.: Sum_{n>=0} [Sum_{d|n} x^(n*d)*(1-x^d)/(1-x)^d].

More terms from R. J. Mathar and Alois P. Heinz, Dec 14 2009

A171634 Number of compositions of n such that the number of parts is divisible by the greatest part.

Original entry on oeis.org

1, 1, 3, 2, 8, 13, 21, 38, 89, 173, 302, 545, 1109, 2309, 4564, 8601, 16188, 31365, 62518, 125813, 251119, 493123, 956437, 1854281, 3633938, 7218166, 14444539, 28868203, 57300450, 112921744, 221760513, 436117749, 861764899, 1711773936

Offset: 1

Vladeta Jovovic, Dec 13 2009

Vaclav Kotesovec, Table of n, a(n) for n = 1..715 (terms 1..250 from Alois P. Heinz)

Cf. A168659, A171632.

b:= proc(n,t,g) option remember; `if`(n=0, `if`(irem(t, g)=0, 1, 0), add(b(n-i, t+1, max(i, g)), i=1..n)) end: a:= n-> b(n,0,0): seq(a(n), n=1..40); # Alois P. Heinz, Dec 15 2009

b[n_, t_, g_] := b[n, t, g] = If[n == 0, If [Mod[t, g] == 0, 1, 0], Sum[b[n - i, t + 1, Max[i, g]], {i, 1, n}]];
a[n_] := b[n, 0, 0];
Array[a, 40] (* Jean-François Alcover, Nov 11 2020, after Alois P. Heinz *)

G.f.: Sum_{n>=0} Sum_{d|n} ((x^(d+1)-x)^n-(x^d-x)^n)/(x-1)^n.

More terms from Alois P. Heinz, Dec 15 2009

A163318 Expansion of g.f.: Product_{k>=1} 1+k*x^k/(1-x^k)^2.

Original entry on oeis.org

1, 1, 4, 8, 19, 36, 76, 142, 272, 496, 900, 1592, 2784, 4792, 8138, 13688, 22703, 37380, 60838, 98310, 157298, 250162, 394332, 618032, 961512, 1487563, 2286610, 3496776, 5316666, 8044598, 12110538, 18147166, 27068692, 40203306, 59459998, 87587428, 128522850

Offset: 0

Vladeta Jovovic, Jul 24 2009

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A006906, A077285, A162506.

b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-1)*(j*i), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 25 2013

terms = 40;
CoefficientList[Product[1 + k x^k/(1 - x^k)^2, {k, 1, terms}] + O[x]^terms, x] (* Jean-François Alcover, Nov 12 2020 *)

cp's OEIS Frontend

User: Vladeta Jovovic

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).

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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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A168655 Number of compositions such that the number of parts is divisible by the first part.

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A171625 Number of compositions of n such that the number of parts is divisible by the smallest part.

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A158615 Expansion of Sum_{n>0} n*n!*x^n/(1-n!*x^n).

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A168173 Number of partitions of n in which the sum of reciprocals of parts is less than 1.

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Mathematica

Extensions

A159034 Inverse Euler transform of A155200.

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A158615 Expansion of Sum_{n>0} nn!x^n/(1-n!*x^n).