A093615 - OEIS

A093615

E.g.f. equals the ratio of two power series, each with triangular exponents of x.

0, 1, -1, 3, -15, 85, -570, 4509, -40804, 414864, -4686570, 58245650, -789691134, 11598605460, -183459343613, 3109122970590, -56203651969935, 1079493501290439, -21953265755518782, 471258656426134701, -10648683969964745520, 252651472831081785300

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OFFSET

0,4

COMMENTS

E.g.f. is asymptotic to 1-1/(2x). Compare to e.g.f. of A093523.

LINKS

Robert Israel, Table of n, a(n) for n = 0..440

FORMULA

E.g.f: T1(x)/T0(x), where T0(x) = sum_{n>=0} x^(n*(n+1)/2)/(n*(n+1)/2)! and T1(x) = sum_{n>=0} x^(n*(n+1)/2+1)/(n*(n+1)/2+1)!; T0(r)=0 at r=-0.8851021553904208809237177147294641529670...

MAPLE

N:= 10: # to get a(0)..a((N+1)*(N+2)/2-1)

T0:= add(x^(n*(n+1)/2)/(n*(n+1)/2)!, n=0..N):

T1:= add(x^(1+n*(n+1)/2)/(1+n*(n+1)/2)!, n=0..N):

S:= series(T1/T0, x, (N+1)*(N+2)/2):

seq(coeff(S, x, n)*n!, n=0..(N+1)*(N+2)/2-1); # Robert Israel, Jan 01 2018

PROG

(PARI) T0(x)=sum(k=0, sqrtint(2*n)+1, x^(k*(k+1)/2)/(k*(k+1)/2)!)

(PARI) T1(x)=sum(k=0, sqrtint(2*n)+1, x^(k*(k+1)/2+1)/(k*(k+1)/2+1)!)

(PARI) a(n)=n!*polcoeff(T1(x)/T0(x)+x*O(x^n), n)

CROSSREFS

Cf. A093523.