A226838 -id:A226838 - cp's OEIS frontend

A226839 E.g.f.: exp( Sum_{n>=1} x^(n*(n+1)/2) / n! ).

1, 1, 1, 4, 13, 31, 271, 1576, 6049, 55693, 573661, 3839716, 36369301, 432793219, 3670898323, 47260464616, 758854978561, 8126729609401, 106290146259289, 1742497711849828, 22974498485218621, 454423040764317031, 8508721270142443351, 120131676428508219784, 2346431431552540513633

Offset: 0

Paul D. Hanna, Jun 19 2013

nonn

E.g.f. may be written as: exp( Sum_{n>=1} Product_{k=1..n} x^k/k ).

Sum_{n>=0} a(n)/n! = e^(e-1) = 5.574941524760880...

			E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 31*x^5/5! + 271*x^6/6! +...
where
log(A(x)) = x + x^3/2! + x^6/3! + x^10/4! + x^15/5! + x^21/6! + x^28/7! +...

Cf. A226838.

{a(n)=n!*polcoeff(exp(sum(m=1,n,x^(m*(m+1)/2)/m!)+x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

a(n) == 1 (mod 3) (conjecture - valid up to n=1024; if true for n>=0, why?).

A226890 E.g.f.: exp( Sum_{n>=1} sigma(n,n) * x^(n^2) / n^n ).

Original entry on oeis.org

1, 1, 1, 1, 31, 151, 451, 1051, 33601, 663601, 5187001, 25905001, 254322751, 10408719751, 128046088171, 920598820051, 29249420054401, 723848667813601, 12441294278905201, 138598703861148241, 4406639731521827551, 93453608310743628151, 1932981245635597160851, 27744052310106087405451

Offset: 0

Paul D. Hanna, Jun 20 2013

nonn

Here sigma(n,n) = A023887(n), the sum of the n-th powers of the divisors of n.

Compare to: exp( Sum_{n>=1} sigma(n)*x^n/n ), the g.f. of the partitions.

			E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + 31*x^4/4! + 151*x^5/5! + 451*x^6/6! +...
where
log(A(x)) = x + 5*x^4/2^2 + 28*x^9/3^3 + 273*x^16/4^4 + 3126*x^25/5^5 + 47450*x^36/6^6 + 823544*x^49/7^7 +...+ A023887(n)*x^(n^2)/n^n +...

Cf. A226838, A023887.

{a(n)=n!*polcoeff(exp(sum(m=1,n,sigma(m,m)*(x^m/m)^m)+x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

a(n) == 1 (mod 30) (conjecture - valid up to n=4000; if true for n>=0, why?).

cp's OEIS Frontend

A226839 E.g.f.: exp( Sum_{n>=1} x^(n*(n+1)/2) / n! ).

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