A158099 Euler transform of square powers of 2: [2,2^4,2^9,...,2^(n^2),...].
1, 2, 19, 548, 66749, 33695574, 68787981855, 563088066184424, 18447871299903970005, 2417888543453357864445634, 1267655436282309648681395304255, 2658458526916981532120588021462151100, 22300750515466692968838881088968809185127601
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 19*x^2 + 548*x^3 + 66749*x^4 +... A(x) = 1/[(1-x)^2*(1-x^2)^(2^4)*(1-x^3)^(2^9)*(1-x^4)^(2^16)*...].
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..35
Programs
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Maple
with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:= etr(n->2^(n^2)): seq(a(n), n=0..15); # Alois P. Heinz, Sep 03 2012
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Mathematica
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; a = etr[Function[{n}, 2^(n^2)]]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *) -
PARI
a(n)=polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^(2^(k^2))),n)
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PARI
{a(n)=polcoeff(exp(sum(m=1,n,sumdiv(m,d,d*2^(d^2))*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, Oct 18 2009
Formula
G.f.: A(x) = 1/Product_{n>=1} (1 - x^n)^(2^(n^2)).
G.f.: exp( Sum_{n>=1} L(n)*x^n/n ) where L(n) = Sum_{d|n} d*2^(d^2). [Paul D. Hanna, Oct 18 2009]