A129483 E.g.f.: A(x) = Product_{n>=0} { exp(x)*[Sum_{k=0..n} (-x)^k/k! ] }.

1, 1, 0, -3, -10, -35, -186, -1162, -6980, -37893, -170170, -420926, 2820168, 58820034, 648789218, 5870211150, 49367781216, 424549221251, 4031944331166, 42858283306334, 485093040406600, 5516989209285204, 60784199053120378, 635030292370785486, 6149124209685347592

Offset

0,4

Comments

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.

Links

Table of n, a(n) for n=0..24.

Formula

Special values: A(1) = 0.

Example

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)! ]*...

Prog

(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)}

Crossrefs

Cf. A129482.

Sequence in context: A151477 A184175 A243563 * A351011 A351009 A119977

Adjacent sequences: A129480 A129481 A129482 * A129484 A129485 A129486

Keyword

sign

Author

Paul D. Hanna, Apr 17 2007

Status

approved