A158099 -id:A158099 - cp's OEIS frontend

A158098 Euler transform of triangular powers of 2: [2,2^3,2^6,...,2^(n(n+1)/2),...].

1, 2, 11, 84, 1217, 35630, 2177587, 273084984, 69282922119, 35324981861270, 36099947418619965, 73859427092428467556, 302379428224074427461199, 2476485356209583877951854650, 40569774298249879934939059013965, 1329309152683489963994724570066550944

Offset: 0

Paul D. Hanna, Mar 20 2009

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..81

Cf. A006125, A158099.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*2^(d*(d+1)/2), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 28 2017

a[n_] := a[n] = If[n==0, 1, Sum[a[n-j] Sum[d 2^(d(d+1)/2), {d, Divisors[j]}], {j, 1, n}]/n];
a /@ Range[0, 20] (* Jean-François Alcover, Nov 20 2020, after Alois P. Heinz *)

a(n)=polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^(2^(k*(k+1)/2))),n)

G.f.: A(x) = 1/Product_{n>=1} (1 - x^n)^(2^(n(n+1)/2)).

A171732 a(n) = Sum_{d|n} d*2^(d^2).

Original entry on oeis.org

2, 34, 1538, 262178, 167772162, 412316861986, 3940649673949186, 147573952589676675106, 21760664753063325144712706, 12676506002282294014967199825954, 29243015907268149203883755326167580674, 267608942382367477698428619271780750388889122

Offset: 1

Paul D. Hanna, Dec 16 2009

nonn

			L.g.f.: L(x) = 2*x + 34*x^2/2 + 1538*x^3/3 + 262178*x^4/4 + ...
exp(L(x)) = 1 + 2*x + 19*x^2 + 548*x^3 + 66749*x^4 + ... (A158099).

Cf. A158099.

a[n_] := DivisorSum[n, # * 2^(#^2) &]; Array[a, 10] (* Amiram Eldar, Aug 18 2023 *)

PARI
```
a(n)=sumdiv(n, d, d*2^(d^2))
```

a(11)-a(12) from Amiram Eldar, Aug 18 2023

cp's OEIS Frontend

A158098 Euler transform of triangular powers of 2: [2,2^3,2^6,...,2^(n(n+1)/2),...].

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula