A032312 "EGJ" (unordered, element, labeled) transform of 2,2,2,2...

1, 2, 4, 14, 48, 162, 826, 3558, 17296, 101714, 529014, 3218118, 21014010, 140974654, 888205714, 6529087674, 52806013456, 375280736754, 2994842092102, 23821110274230, 217847892367318, 1894959770821614, 16188955616322394, 142246084665611010, 1376483692715941594

Offset: 0

Author

Christian G. Bower

Keywords

• nonn

Comments

From Peter Bala, Sep 05 2022: (Start)

Conjecture: the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Cf. A007837.

Equivalently, the expansion of exp( Sum_{n >= 1} a(n)^x^n/n ) = 1 + 2*x + 4*x^2 + 10*x^3 + 28*x^4 + 82*x^5 + 293*x^6 + ... has integer coefficients. Cf. A168268. (End)

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• C. G. Bower, Transforms (2)

Crossrefs

Cf. A007837, A168268.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Product[(1+x^k/k!)^2,{k,nn}],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 07 2019 *)

PARI

seq(n)={Vec(serlaplace(prod(k=1, n, (1 + x^k/k! + O(x*x^n))^2)))} \\ Andrew Howroyd, Sep 11 2018

Formula

E.g.f: Product_{k > 0} (1 + x^k/k!)^2. - Andrew Howroyd, Sep 11 2018

Extensions

a(0)=1 prepended and terms a(22) and beyond from Andrew Howroyd, Sep 11 2018