A007838 -id:A007838 - cp's OEIS frontend

A182966 E.g.f.: A(x) = Product_{n>=1} (1 + 3*x^n/n)^n.

1, 3, 6, 72, 342, 3330, 36720, 366660, 4974480, 67178160, 1043189280, 16836906240, 303306806880, 5705780240160, 114832957599360, 2475901844095680, 55754442891987840, 1331875774475326080, 33292197644365820160

Offset: 0

Paul D. Hanna, Dec 19 2010

nonn

			E.g.f.: A(x) = 1 + 3*x + 6*x^2/2! + 72*x^3/3! + 342*x^4/4! +...
A(x) = (1+3x)*(1+3x^2/2)^2*(1+3x^3/3)^3*(1+3x^4/4)^4*(1+3x^5/5)^5*...

Vaclav Kotesovec, Table of n, a(n) for n = 0..438

Cf. A181541, A182965, A182967, A007838.

nmax = 20; CoefficientList[Series[Product[(1 + 3*x^k/k)^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 07 2020 *)

{a(n,k=3)=n!*polcoeff(prod(m=1,n,(1+k*x^m/m+x*O(x^n))^m),n)}

A182967 E.g.f.: A(x) = Product_{n>=1} (1 + 4*x^n/n)^n.

Original entry on oeis.org

1, 4, 8, 120, 576, 6240, 75840, 772800, 11585280, 163914240, 2694558720, 45947489280, 876665180160, 17329568256000, 364677585592320, 8306018798837760, 195321474697789440, 4892032896606535680

Offset: 0

Paul D. Hanna, Dec 19 2010

nonn

			E.g.f.: A(x) = 1 + 4*x + 8*x^2/2! + 120*x^3/3! + 576*x^4/4! +...
A(x) = (1+4x)*(1+4x^2/2)^2*(1+4x^3/3)^3*(1+4x^4/4)^4*(1+4x^5/5)^5*...

Vaclav Kotesovec, Table of n, a(n) for n = 0..436

Cf. A181541, A182965, A182966, A007838.

With[{nn=20},CoefficientList[Series[Product[(1+4 x^n/n)^n,{n,nn}],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 11 2020 *)

{a(n,k=4)=n!*polcoeff(prod(m=1,n,(1+k*x^m/m+x*O(x^n))^m),n)}

A308338 Expansion of e.g.f. exp(-1 + Product_{k>=1} (1 + x^k/k)).

Original entry on oeis.org

1, 1, 2, 9, 44, 270, 2064, 17682, 171296, 1867968, 22470840, 294493320, 4195969392, 64416698112, 1059685905264, 18609306423120, 347179119075840, 6855335163907200, 142889687354283264, 3133647091691585280, 72124075333003155840, 1738384773846440146560

Offset: 0

Ilya Gutkovskiy, May 20 2019

nonn

Cf. A007838, A308336, A308337.

nmax = 21; CoefficientList[Series[Exp[Product[(1 + x^k/k), {k, 1, nmax}] - 1], {x, 0, nmax}], x] Range[0, nmax]!

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1)*A007838(k)*a(n-k).

A317166 Number of permutations of [n] with distinct lengths of increasing runs.

Original entry on oeis.org

1, 1, 1, 5, 7, 27, 241, 505, 1975, 10241, 188743, 460545, 2323679, 10836141, 85023209, 2734858573, 8010483015, 45714797671, 243112435345, 1632532938001, 15831051353773, 892173483721887, 2978105991739613, 19855526019022967, 113487352591708591

Offset: 0

Alois P. Heinz, Jul 23 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..66

Cf. A000217, A007838, A317165.

g:= (n, s)-> `if`(n in s, 0, 1):
b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
      `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
       , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
    end:
a:= n-> b(n, 0$2, {}):
seq(a(n), n=0..24);

g[n_, s_] := If[MemberQ[s, n], 0, 1];
b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
     If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
     {j, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, o}]];
a[n_] := b[n, 0, 0, {}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Sep 01 2021, after Alois P. Heinz *)

a(A000217(n)) = A317165(n).

A318694 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(ij)/(ij)).

Original entry on oeis.org

1, 1, 2, 10, 40, 248, 1868, 14516, 131920, 1409040, 15697872, 191687472, 2663239104, 37878672960, 582357866400, 9898540886880, 172534018584960, 3192686545714560, 63844374067107840, 1309775114921541120, 28512040933544970240, 656888836504576112640, 15495311684125737031680

Offset: 0

Ilya Gutkovskiy, Aug 31 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..447

Cf. A000005, A007838, A107742, A181541, A318416, A318693.

seq(n!*coeff(series(mul((1+x^k/k)^tau(k),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019

nmax = 22; CoefficientList[Series[Product[Product[(1 + x^(i j)/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + x^k/k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-d)^(1 - k/d) DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-d)^(1 - k/d) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

E.g.f.: Product_{k>=1} (1 + x^k/k)^tau(k), where tau = number of divisors (A000005).

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-d)^(1-k/d)*tau(d) ) * x^k/k).

A326855 E.g.f.: Product_{k>=1} (1 + x^(4k-1) / (4k-1)).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 720, 0, 0, 172800, 3628800, 0, 0, 2641766400, 87178291200, 0, 0, 225422681702400, 6402373705728000, 0, 221172909834240000, 30424079849619456000, 1124000727777607680000, 0, 49241936645495193600000, 11321261082950211993600000

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
Vaclav Kotesovec, Graph - the asymptotic ratio (40000 terms)

Cf. A007838, A088994, A326779, A326856, A326857.

nmax = 30; CoefficientList[Series[Product[(1+x^(4*k-1)/(4*k-1)), {k, 1, Floor[nmax/4]+1}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ Pi * n! / (exp(gamma/4) * Gamma(1/4)^2 * n^(3/4)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

A326856 E.g.f.: Product_{k>=1} (1 + x^(4k-3) / (4k-3)).

Original entry on oeis.org

1, 1, 0, 0, 0, 24, 144, 0, 0, 40320, 403200, 0, 0, 479001600, 8643317760, 29059430400, 0, 20922789888000, 475108274995200, 1871463083212800, 0, 2432902008176640000, 76354225980899328000, 525098781423304704000, 0, 620448401733239439360000

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

In general, if c > 0, mod(d,c) <> 0 and e.g.f. = Product_{k>=1} (1 + x^(c*k+d) / (c*k+d)), then a(n) ~ n! * Gamma(1 + d/c) / (c^(1/c) * exp(gamma/c) * Gamma(1/c) * Gamma(1 + (d+1)/c) * n^(1 - 1/c)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
Vaclav Kotesovec, Graph - the asymptotic ratio (40000 terms)

Cf. A007838, A088994, A326780, A326855.

nmax = 30; CoefficientList[Series[Product[(1+x^(4*k-3)/(4*k-3)), {k, 1, Floor[nmax/4]+1}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ n! / (sqrt(2*Pi) * exp(gamma/4) * n^(3/4)), where gamma is the Euler-Mascheroni constant A001620.

A326858 E.g.f.: Product_{k>=1} (1 + x^(3k-2) / (3k-2)).

Original entry on oeis.org

1, 1, 0, 0, 6, 30, 0, 720, 5760, 0, 362880, 5417280, 17107200, 479001600, 8885479680, 32691859200, 1307674368000, 34151856076800, 214585052774400, 6402373705728000, 192796754895360000, 1542202547010048000, 55105230485200896000, 1944933030182596608000

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A007838, A088994, A326756, A326857.

nmax = 30; CoefficientList[Series[Product[(1+x^(3*k-2)/(3*k-2)), {k, 1, Floor[nmax/3]+1}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ n! / (3^(1/3) * Gamma(2/3) * exp(gamma/3) * n^(2/3)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

A326865 G.f.: Product_{k>=1} (1 + x^k/k^3) = Sum_{n>=0} a(n)*x^n/n!^3.

Original entry on oeis.org

1, 1, 1, 35, 728, 48824, 7170984, 1418111064, 479963197440, 235727037775872, 170423013422592000, 163854260184125952000, 214343327259234349056000, 360795240553638133592064000, 778954481701636984110452736000, 2095759092922096320907078496256000

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..182

Cf. A007838, A249593, A326864.

b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 27 2023

nmax = 20; CoefficientList[Series[Product[(1+x^k/k^3), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!^3

a(n) ~ c * (n-1)!^3, where c = A073017 = Product_{k>=1} (1 + 1/k^3) = cosh(sqrt(3)*Pi/2)/Pi = 2.428189792098870328736...

A335637 Expansion of e.g.f. Product_{k>0} (1 + sin(x)^k / k).

Original entry on oeis.org

1, 1, 1, 4, 10, 25, 210, 978, 2336, 25265, 361424, 1557752, -1098528, 140915385, 2093367328, 10484632486, 133131785728, -1343478380255, -8738565516288, 1790935681747980, 3245598828836864, -592809746388403495, 6832010190766985216, 179327221659613996634, -5310378915096702812160

Offset: 0

Seiichi Manyama, Oct 03 2020

sign

Cf. A007838, A335629, A335635, A335638, A335644.

max = 24; Range[0, max]! * CoefficientList[Series[Product[1 + Sin[x]^k/k, {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 03 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1+sin(x)^k/k)))

N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, (-1)^(i+1)*sin(x)^(i*j)/(i*j^i))))))

E.g.f.: exp( Sum_{i>0} Sum_{j>0} (-1)^(i+1)*sin(x)^(i*j)/(i*j^i) ).

cp's OEIS Frontend

A182966 E.g.f.: A(x) = Product_{n>=1} (1 + 3*x^n/n)^n.

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A182967 E.g.f.: A(x) = Product_{n>=1} (1 + 4*x^n/n)^n.

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A308338 Expansion of e.g.f. exp(-1 + Product_{k>=1} (1 + x^k/k)).

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A317166 Number of permutations of [n] with distinct lengths of increasing runs.

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A318694 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(i*j)/(i*j)).

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A326855 E.g.f.: Product_{k>=1} (1 + x^(4*k-1) / (4*k-1)).

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A326856 E.g.f.: Product_{k>=1} (1 + x^(4*k-3) / (4*k-3)).

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A326858 E.g.f.: Product_{k>=1} (1 + x^(3*k-2) / (3*k-2)).

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A326865 G.f.: Product_{k>=1} (1 + x^k/k^3) = Sum_{n>=0} a(n)*x^n/n!^3.

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A318694 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(ij)/(ij)).

A326855 E.g.f.: Product_{k>=1} (1 + x^(4k-1) / (4k-1)).

A326856 E.g.f.: Product_{k>=1} (1 + x^(4k-3) / (4k-3)).

A326858 E.g.f.: Product_{k>=1} (1 + x^(3k-2) / (3k-2)).