A143739 G.f. A(x) satisfies: A(x) = (1-x)^3*A(x)^2 - x^2*A'(x).

1, 3, 9, 28, 90, 300, 1051, 3975, 16971, 86584, 550560, 4354308, 41245021, 448722207, 5443128597, 72294557416, 1039558994214, 16059538853232, 264996063891607, 4648786414414347, 86363450200625247, 1693336666564186012

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) ~ (n-1)! / 2. - Vaclav Kotesovec, Feb 22 2014

for n > 1, a(n) = (n+1)*(n+2)/2 + (n-1)*a(n-1) + Sum_{k=1..n-1}(-a(k) + (k+1)*(k+2)/2)*a(n-k) + (k+1)*(k+2)*(n-1-k)*a(n-1 - k)/2. - Tani Akinari, Nov 29 2017

Example

G.f.: A(x) = 1 + 3*x + 9*x^2 + 28*x^3 + 90*x^4 + 300*x^5 + 1051*x^6 + ...
A(x)^2 = 1 + 6*x + 27*x^2 + 110*x^3 + 429*x^4 + 1644*x^5 + 6306*x^6 + ...
(1-x)^3*A(x)^2 = 1 + 3*x + 12*x^2 + 46*x^3 + 174*x^4 + 660*x^5 + ...
x^2*A'(x) = 3*x^2 + 18*x^3 + 84*x^4 + 360*x^5 + 1500*x^6 + 6306*x^7 + ...

Mathematica

Table[Coefficient[Series[2*E^x/(E^x*(2 + (-5 + 1/x)*(1/x)) - ExpIntegralEi[x]), {x, Infinity, 20}], x, -n], {n, 0, 20}] (* Vaclav Kotesovec, Feb 22 2014 *)

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1+x^2*deriv(A)/A)/(1-x)^3); polcoeff(A, n)}
(Maxima) a[n]:=if n<2 then 2*n+1 else (n+1)*(n+2)/2+(n-1)*a[n-1]+sum((-a[k]+(k+1)*(k+2)/2)*a[n-k]+(k+1)*(k+2)*(n-1-k)*a[n-1-k]/2, k, 1, n-1);
makelist(a[n], n, 0, 100); /* Tani Akinari, Nov 29 2017 */

Crossrefs

Cf. A137553 (variant).

Sequence in context: A370074 A071724 A000245 * A189940 A047047 A071744

Adjacent sequences: A143736 A143737 A143738 * A143740 A143741 A143742

Keyword

nonn

Author

Paul D. Hanna, Sep 05 2008

Status

approved