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

0,2

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)

Formula

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

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 *)

Prog

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

Crossrefs

Cf. A007837, A168268.

Sequence in context: A006443 A152103 A102879 * A032222 A212268 A092665

Adjacent sequences: A032309 A032310 A032311 * A032313 A032314 A032315

Keyword

nonn

Author

Christian G. Bower

Extensions

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

Status

approved