A265952 -id:A265952 - cp's OEIS frontend

A265953 E.g.f.: Product_{k>=1} 1/(1 - exp(x)*x^k).

1, 1, 6, 39, 328, 3305, 39396, 536053, 8210784, 139670721, 2612934820, 53260680341, 1175587507392, 27929705129521, 710678763809028, 19284199100275845, 555961318128936256, 16972543570002866945, 547046699544108738756, 18566047855851466092949

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..408

Cf. A030178, A030797, A265952.

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

a(n) ~ c * n! / LambertW(1)^n, where c = 1/(1 + LambertW(1)) * Product_{j>=1} 1/(1 - LambertW(1)^j) = 3.40413121452412914124892504613759007312040569..., LambertW(1) = A030178.

A347006 E.g.f.: Product_{k>=1} (1 + exp(x) * x^k / k!).

Original entry on oeis.org

1, 1, 3, 10, 43, 206, 1044, 5909, 38371, 272314, 1995208, 14869889, 115433344, 965259881, 8773348601, 84608514095, 837220780691, 8334354200226, 83498917650084, 855936118936073, 9180736840445788, 104439240481045949, 1253608634906635901

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

a(n) is the number of words of length n over an infinite alphabet such that for any letter k appearing within a word, exactly k occurrences of that letter are marked. - John Tyler Rascoe, Jul 16 2025

			a(3) = 10 counts: (1#,1,1), (1,1#,1), (1,1,1#), (1#,2#,2#), (2#,1#,2#), (2#,2#,1#), (2#,2#,2), (2#,2,2#), (2,2#,2#), (3#,3#,3#). - _John Tyler Rascoe_, Jul 16 2025

Alois P. Heinz, Table of n, a(n) for n = 0..612 (first 101 terms from John Tyler Rascoe)

Cf. A007837, A265952, A305547, A347005.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-j, min(n-j, i-1))/i!/(j-i)!, j=i..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 17 2025

nmax = 22; CoefficientList[Series[Product[(1 + Exp[x] x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

C_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + exp(x)*x^k/k!)))} \\ John Tyler Rascoe, Jul 16 2025

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

E.g.f.: Product_{k>=1} (1 + Sum_{j>=k} binomial(j,k) * x^j / j!).

A302399 Expansion of e.g.f. Product_{k>=1} 1/(1 - exp(x)*x^k)^k.

Original entry on oeis.org

1, 1, 8, 63, 628, 7405, 103266, 1630195, 28812344, 561715353, 11971270270, 276322667071, 6867229990644, 182651988444133, 5174629835814362, 155498722020145995, 4938797154614179696, 165259917542803746097, 5809661798192528407542, 214032701720169039806551, 8244827039453943163648940

Offset: 0

Ilya Gutkovskiy, Apr 07 2018

nonn

			Product_{k>=1} 1/(1 - exp(x)*x^k)^k = 1 + x/1! + 8*x^2/2! + 63*x^3/3! + 628*x^4/4! + 7405*x^5/5! + 103266*x^6/6! + ...

Cf. A265952, A265953.

a:=series(mul(1/(1-exp(x)*x^k)^k,k=1..100),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019

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

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

a(n) ~ c * n! / LambertW(1)^n, where c = 1/(1 + LambertW(1)) * Product_{j>=1} 1/(1 - LambertW(1)^j)^(j+1) = 115.50749040505570853455997830821388214033876679679... - Vaclav Kotesovec, Apr 07 2018

A322613 Expansion of e.g.f. Product_{k>=1} (1 - log(1 - x)*x^k).

Original entry on oeis.org

1, 0, 2, 9, 44, 370, 3084, 32088, 336384, 4407408, 59113440, 896773680, 14403234240, 250498939392, 4625127900288, 92232410538240, 1925532322237440, 42709138254167040, 997150775080043520, 24416143271431649280, 626110124433676185600, 16824255461119247339520, 471015493365385119191040

Offset: 0

Ilya Gutkovskiy, Dec 20 2018

nonn

Cf. A126348, A265952, A322612.

seq(coeff(series(factorial(n)*mul((1-log(1-x)*x^k),k=1..n),x,n+1), x, n), n = 0 .. 22); # Muniru A Asiru, Dec 21 2018

nmax = 22; CoefficientList[Series[Product[(1 - Log[1 - x] x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(d + 1) Log[1/(1 - x)]^d/d, {d, Divisors[k]}] x^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

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

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A265953 E.g.f.: Product_{k>=1} 1/(1 - exp(x)*x^k).

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A347006 E.g.f.: Product_{k>=1} (1 + exp(x) * x^k / k!).

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A302399 Expansion of e.g.f. Product_{k>=1} 1/(1 - exp(x)*x^k)^k.

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A322613 Expansion of e.g.f. Product_{k>=1} (1 - log(1 - x)*x^k).

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