A167006 -id:A167006 - cp's OEIS frontend

A227467 E.g.f.: exp( Sum_{n>=1} (1+x)^(n^2) * x^n/n ).

1, 1, 4, 24, 252, 3660, 73560, 1921080, 63411600, 2574406800, 125747475840, 7258472907840, 487590023511360, 37629962101892160, 3299990581104497280, 325758967714868688000, 35904380354917794720000, 4387164775718671231084800, 590610815931660911894707200, 87118296156852814044256665600

Offset: 0

Paul D. Hanna, Aug 24 2013

nonn

Compare the definition to: exp( Sum_{n>=1} (1+y)^(n^2) * x^n/n ), which yields an integer series whenever y is an integer.

Note that exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * x^k ) yields an integer series (A206830).

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 24*x^3/3! + 252*x^4/4! + 3660*x^5/5! +...
where, by definition,
log(A(x)) = (1+x)*x + (1+x)^4*x^2/2 + (1+x)^9*x^3/3 + (1+x)^16*x^4/4 + (1+x)^25*x^5/5+ (1+x)^36*x^6/6+ (1+x)^49*x^7/7 +...

Cf. A206830, A167006.

{a(n)=n!*polcoeff(exp(sum(m=1, n, (1+x)^(m^2)*x^m/m)+x*O(x^n)), n)}
for(n=0,25,print1(a(n),", "))

A207139 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k) * binomial(n^2,k^2) ).

Original entry on oeis.org

1, 2, 7, 147, 14481, 6183605, 19196862399, 206667738393577, 6727813723143519624, 1368162090055314881480420, 1237384559488983889303951699285, 3014186760620644058660289396656407831, 34123084437870355957570087446546456971276065

Offset: 0

Paul D. Hanna, Feb 15 2012

nonn

The logarithmic derivative yields A207140.

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 147*x^3 + 14481*x^4 + 6183605*x^5 +...
where the logarithm of the g.f. equals the l.g.f. of A207140:
log(A(x)) = x + 2*x^2/2 + 10*x^3/3 + 407*x^4/4 + 56746*x^5/5 +...

Cf. A207140 (log), A206850, A207135, A207137, A167006.

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)*binomial(m^2,k^2))+x*O(x^n))),n)}
for(n=0,16,print1(a(n),", "))

cp's OEIS Frontend

A227467 E.g.f.: exp( Sum_{n>=1} (1+x)^(n^2) * x^n/n ).

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