A007840 -id:A007840 - cp's OEIS frontend

A009566 Expansion of e.g.f. sinh(log(1+log(1+x))).

0, 1, -2, 8, -47, 359, -3349, 36756, -462972, 6574704, -103861896, 1805930928, -34267701720, 704555489832, -15601914340440, 370195510608192, -9369769657984128, 251979258929744448, -7175127496412091456

Offset: 0

R. H. Hardin

G. C. Greubel, Table of n, a(n) for n = 0..415

CoefficientList[Series[(Log[1 + x]*(2 + Log[1 + x]))/(2*(1 + Log[1 + x]) ), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 23 2015 *)
With[{nmax = 50}, CoefficientList[Series[Sinh[Log[1 + Log[1 + x]]], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jan 21 2018 *)

my(x='x+O('x^30)); concat([0], Vec(serlaplace(sinh(log(1+log(1+x)))))) \\ G. C. Greubel, Jan 21 2018

a(n) ~ n! * (-1)^(n+1) * exp(n) / (2 * (exp(1)-1)^(n+1)). - Vaclav Kotesovec, Jan 23 2015

a(n) = (-1)^(n-1)*((n-1)! + A007840(n))/2 for n > 0. - Velin Yanev, May 17 2024

Extended with signs by Olivier Gérard, Mar 15 1997

A218817 Number of rooted factorizations of n-permutations into ordered cycles.

Original entry on oeis.org

0, 1, 6, 42, 352, 3470, 39468, 509544, 7367232, 117981792, 2073609120, 39690563616, 821945839680, 18312215714832, 436766423241120, 11104557643877760, 299811706265604096, 8566939116183215232, 258298187497129564416, 8195130059917806607104, 272936837532680503188480

Offset: 0

Geoffrey Critzer, Nov 06 2012

nonn

Linearly arrange the cycles over all permutations of {1,2,...,n} (these are called alignments in [Flajolet and Sedgewick]) then select a root.

Alois P. Heinz, Table of n, a(n) for n = 0..150
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 119.

b:= proc(n) b(n):= n!*`if`(n=0, 1, add(b(k)/(k!*(n-k)), k=0..n-1)) end:
a:= n-> n*b(n):
seq(a(n), n=0..20); # Alois P. Heinz, Nov 06 2012

nn=20;a=Log[1/(1-x)];Range[0,nn]!CoefficientList[Series[x D[1/(1-a),x] ,{x,0,nn}],x]

E.g.f.: x/( (1-x)*(1 - log(1/(1-x)))^2 ).

a(n) = n*A007840(n).

A303000 a(n) = permanent of the n X n matrix with entries a(i, i) = i^2 and a(i, j) = 1 elsewhere.

Original entry on oeis.org

1, 5, 52, 918, 24630, 934938, 47736048, 3157054776, 262661665176, 26857133054424, 3311299323349920, 484541686800059760, 83031688670103506160, 16472545369548670950480, 3746065113561656467249920, 968109978211279792380074880, 282158259444905145777416119680

Offset: 1

Vaclav Kotesovec, Apr 17 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..36

Cf. A007840, A010791, A067550, A118714, A303001.

Table[Permanent[DiagonalMatrix[Table[i^2-1, {i, 1, n}]] + 1], {n, 1, 20}]

{a(n) = matpermanent(matrix(n, n, i, j, if(i==j, i^2, 1)))}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Dec 20 2018

A303001 a(n) = permanent of the n X n matrix with entries a(i, i) = i*(i+1)/2 and a(i, j) = 1 elsewhere.

Original entry on oeis.org

1, 4, 30, 356, 6106, 142760, 4363848, 168986136, 8087082144, 468807362736, 32379640476000, 2627735592279600, 247610398718738640, 26815386224063189760, 3307855985755600598400, 461149884030679844958720, 72151124506962747825006720, 12590610689213961622942752000

Offset: 1

Vaclav Kotesovec, Apr 17 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..36

Cf. A007840, A010791, A067550, A118714, A303000.

Table[Permanent[DiagonalMatrix[Table[i*(i+1)/2-1, {i, 1, n}]] + 1], {n, 1, 20}]

{a(n) = matpermanent(matrix(n, n, i, j, if(i==j, i*(i+1)/2, 1)))}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Dec 20 2018

A318617 a(n) is the number of rooted labeled forests on n nodes that avoid the patterns 213, 231, and 312.

Original entry on oeis.org

1, 1, 3, 13, 73, 503, 4107, 38773, 415589, 4986715, 66238503, 965102769, 15306905817, 262567910999, 4844199561787, 95660129298709, 2013392566243565, 44997370759528091, 1064283567185090791, 26560710262784693097, 697529916604465424553

Offset: 0

Kassie Archer, Aug 30 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
K. Anders and K. Archer, Rooted forests that avoid sets of permutations, arXiv:1607.03046 [math.CO], 2017.

Cf. A000272, A318618, A007840, A000671, A000262.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, k-1] (r-1)! a[n-k] a[k-r], {k, 1, n}, {r, 1, k}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Sep 13 2018 *)

seq(n)={my(v=vector(n+1)); v[1]=1; for(n=1, n, v[n+1]=sum(k=1, n, sum(r=1, k, binomial(n-1,k-1)*(r-1)!*v[n-k+1]*v[k-r+1]))); v} \\ Andrew Howroyd, Aug 30 2018

a(n) = Sum_{k=1..n} Sum_{r=1..k} binomial(n-1,k-1)*(r-1)!*a(n-k)*a(k-r) for n>0, a(0)=1.

A318618 a(n) is the number of rooted forests on n nodes that avoid the patterns 321, 2143, and 3142.

Original entry on oeis.org

1, 1, 3, 15, 102, 870, 8910, 106470, 1454040, 22339800, 381364200, 7161323400, 146701724400, 3255661609200, 77808668137200, 1992415575150000, 54420258228336000, 1579320261543024000, 48529229906613456000, 1574046971727454224000, 53741325186841612320000

Offset: 0

Kassie Archer, Aug 30 2018

nonn

a(n) is the number of rooted labeled forests on n nodes so that along any path from the root to a vertex, there is at most one descent.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Peter J. Taylor, Improved formulae for A318618

Cf. A000272, A318617, A007840, A000671, A000262.

a[n_] := n! + Sum[n! 2^-j Binomial[n-k-1, j-1] Binomial[k, j], {k, 1, n}, {j, 1, Min[k, n-k]}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Sep 13 2018 *)

a(n) = {n! + sum(k=1, n, sum(j=1, min(k, n-k), n!/(2^j)*binomial(n-k-1,j-1)*binomial(k,j)))} \\ Andrew Howroyd, Aug 31 2018

a(n) = n! + Sum_{k=1..n} Sum_{j=1..min(k, n-k)} (n!/2^j)*binomial(n-k-1, j-1)*binomial(k, j).
From Peter J. Taylor, Jul 03 2025: (Start)
E.g.f.: -2*(x-1)/(x^2-4*x+2).
a(n) = n! * Sum_{j=0..n/2} binomial(n, 2*j)/2^j
a(n) = 2*n*a(n-1) - n*(n-1)/2*a(n-2).
a(n) ~ (1+sqrt(1/2))^n*n!/2. (End)

A330499 Expansion of e.g.f. Sum_{k>=1} log(1 + log(1/(1 - x))^k).

Original entry on oeis.org

0, 1, 2, 13, 71, 558, 5344, 60926, 766898, 10759096, 168848256, 2947203048, 56368708824, 1165246323408, 25802649445728, 609940593443952, 15377212949988624, 412827548455415040, 11764577341464710016, 354392697960438122880, 11237993013428254071936

Offset: 0

Vaclav Kotesovec, Dec 16 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A007840, A330354, A330388, A330494, A330498.

nmax = 20; CoefficientList[Series[Sum[Log[1+Log[1/(1-x)]^k], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ n! * c / (1 - exp(-1))^n, where c = 0.478656...

A331339 E.g.f.: 1 / (1 + log(1 - x - x^2)).

Original entry on oeis.org

1, 1, 5, 32, 292, 3294, 44918, 714468, 13002456, 266275200, 6060498672, 151750887936, 4145522908272, 122690391196944, 3910569680464848, 133549150323123744, 4864927063250290176, 188297220693251438208, 7716800776602560577408

Offset: 0

Ilya Gutkovskiy, Jan 14 2020

nonn

Cf. A000204, A007840, A039647.

A331339 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        add(binomial(n,k)*(k-1)!*A000204(k)*procname(n-k),k=1..n) ;
    end if;
end proc:
seq(A331339(n),n=0..42) ; # R. J. Mathar, Aug 20 2021

nmax = 18; CoefficientList[Series[1/(1 + Log[1 - x - x^2]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (k - 1)! LucasL[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

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

a(n) ~ n! * 2^(n+1) * exp(n/2) / (sqrt(5*exp(1) - 4) * (sqrt(5*exp(1) - 4) - exp(1/2))^(n+1)). - Vaclav Kotesovec, Jan 26 2020

A331340 a(n) = n! * [x^n] 1 / (1 + Sum_{k=1..n} log(1 - k*x)).

Original entry on oeis.org

1, 1, 23, 1872, 371524, 147316050, 102823452318, 115685840003328, 196669439127051840, 480847207762313690400, 1626231663646322798946000, 7372321556702072183715972096, 43653032698484678876818157764224, 330351436922959495109028135649934640

Offset: 0

Ilya Gutkovskiy, Jan 14 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..167

Cf. A007840, A319508, A319509, A331341.

Table[n! SeriesCoefficient[1/(1 + Sum[Log[1 - k x], {k, 1, n}]), {x, 0, n}], {n, 0, 13}]
Table[n! SeriesCoefficient[1/(1 + Log[Sum[StirlingS1[n + 1, n - k + 1] x^k, {k, 0, n}]]), {x, 0, n}], {n, 0, 13}]

a(n) = n! * [x^n] 1 / (1 + log(Sum_{k=0..n} Stirling1(n+1,n-k+1) * x^k)).

a(n) ~ sqrt(Pi) * n^(3*n + 1/2) / (2^(n - 1/2) * exp(n - 5/3)). - Vaclav Kotesovec, Jan 28 2020

A343685 a(0) = 1; a(n) = 2 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 3, 19, 182, 2328, 37234, 714674, 16004064, 409587144, 11792756640, 377261048592, 13275818803488, 509646721402032, 21195285059025648, 949279217570464944, 45552467588773815744, 2331624264279599225088, 126804353256754734370176, 7301857349340031590836352, 443826900013575494233057536

Offset: 0

Ilya Gutkovskiy, Apr 26 2021

nonn

Cf. A007840, A010842, A052820, A343686, A343687.

a[0] = 1; a[n_] := a[n] = 2 n a[n - 1] + Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[1/(1 - 2 x + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: 1 / (1 - 2*x + log(1 - x)).

a(n) ~ n! / ((2/c + 1 - c) * (1 - c/2)^n), where c = LambertW(2*exp(1)) = 1.3748225281836233816178373171119... - Vaclav Kotesovec, Apr 26 2021

cp's OEIS Frontend

A009566 Expansion of e.g.f. sinh(log(1+log(1+x))).

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A218817 Number of rooted factorizations of n-permutations into ordered cycles.

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A303000 a(n) = permanent of the n X n matrix with entries a(i, i) = i^2 and a(i, j) = 1 elsewhere.

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A303001 a(n) = permanent of the n X n matrix with entries a(i, i) = i*(i+1)/2 and a(i, j) = 1 elsewhere.

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A318617 a(n) is the number of rooted labeled forests on n nodes that avoid the patterns 213, 231, and 312.

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A318618 a(n) is the number of rooted forests on n nodes that avoid the patterns 321, 2143, and 3142.

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A330499 Expansion of e.g.f. Sum_{k>=1} log(1 + log(1/(1 - x))^k).

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A331339 E.g.f.: 1 / (1 + log(1 - x - x^2)).

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A331340 a(n) = n! * [x^n] 1 / (1 + Sum_{k=1..n} log(1 - k*x)).

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