0,4

Sum of series at x=1 converges to zero: 0 = 1 +1 +0/2! -3/3! -10/4! -35/5! +... G.f. for A129482 is closely related.

Special values: A(1) = 0.

E.g.f.: A(x) = 1 +x +0x^2/2! -3x^3/3! -10x^4/4! -35x^5/5! -186x^6/6!

-1162x^7/7! -6980x^8/8! -37893x^9/9! -170170x^10/10! -420926x^11/11! +...

Product formula is illustrated by:

A(x) = [exp(x)*(1)]*[exp(x)*(1 - x)]*[exp(x)*(1 - x + x^2/2!)]*

[exp(x)*(1 - x + x^2/2! - x^3/3!)]*

[exp(x)*(1 - x + x^2/2! - x^3/3! + x^4/4!)]*

[exp(x)*(1 - x + x^2/2! - x^3/3! + x^4/4! - x^5/5!)]*...*

[exp(x)*(Sum_{k=0..n} (-x)^k/k!) ]*...

Equivalently:

A(x) = [1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! +...]*

[1 - x^2/2! - 2x^3/3! - 3x^4/4! - 4x^5/5! - 5x^6/6! -...]*

[1 + x^3/3! + 3x^4/4! + 6x^5/5! + 10x^6/6! + 15x^7/7! +...]*

[1 - x^4/4! - 4x^5/5! - 10x^6/6! - 20x^7/7! - 35x^8/8! -...]*

[1 + x^5/5! + 5x^6/6! + 15x^7/7! + 35x^8/8! + 70x^9/9! +...]*...*

[1 + (-1)^n*Sum_{k>=0} C(n+k-1,n)*x^(n+k)/(n+k)! ]*...

(PARI) {a(n)=n!*polcoeff(prod(k=0, n, exp(x+x*O(x^n))*sum(j=0, k, (-x)^j/j!)), n)} /* Alternate Formula: */ {a(n)=n!*polcoeff(prod(k=0, n, 1+(-1)^k*sum(i=1, n-k+1, binomial(k+i-1, k)*x^(k+i)/(k+i)! +x*O(x^n))), n)}

Cf. A129482.

Adjacent sequences: A129480 A129481 A129482 this_sequence A129484 A129485 A129486

Sequence in context: A084781 A008984 A097148 this_sequence A075890 A119977 A075316

sign

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 17 2007