cp's OEIS Frontend

A202826 E.g.f.: exp( 1/(1-x)^3 - 1 ).

%I A202826 #28 Nov 22 2018 07:28:16
%S A202826 1,3,21,195,2241,30483,476469,8383203,163532385,3496040163,
%T A202826 81159271029,2030708891907,54427341596769,1554460972941555,
%U A202826 47097454520401749,1507969940021725347,50850987639474121281,1800630391669594010307,66775808799868618561365
%N A202826 E.g.f.: exp( 1/(1-x)^3 - 1 ).
%H A202826 Seiichi Manyama, <a href="/A202826/b202826.txt">Table of n, a(n) for n = 0..420</a> (terms 0..200 from Vincenzo Librandi)
%F A202826 a(n) = Sum_{k=0..n} Stirling1(n, k) * Bell(k) * (-1)^(n-k) * 3^k.
%F A202826 a(n) ~ n! * 1/2*3^(1/8)*exp(sqrt(3*n)/2 -3/4 + (3*n)^(1/4)*(4/3*sqrt(n) + 5/24*sqrt(3)) )/(sqrt(2*Pi)*n^(5/8)) * (1 + 871/2560*(3/n)^(1/4)). - _Vaclav Kotesovec_, Feb 12 2013
%F A202826 a(n+4) - (4*n+15)*a(n+3) + 6*(n+2)*(n+3)*a(n+2) - 4*(n+1)*(n+2)+(n+3)*a(n+1) + n*(n+1)*(n+2)*(n+3)*a(n) = 0. - _Emanuele Munarini_, Sep 01 2017
%e A202826 E.g.f.: A(x) = 1 + 3*x + 21*x^2/2! + 195*x^3/3! + 2241*x^4/4! +...
%e A202826 where
%e A202826 log(A(x)) = 3*x + 6*x^2 + 10*x^3 + 15*x^4 + 21*x^5 + 28*x^6 +...
%t A202826 CoefficientList[Series[E^(1/(1-x)^3-1), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Feb 12 2013 *)
%t A202826 Table[Sum[Abs[StirlingS1[n, k]] 3^k BellB[k], {k, 0, n}], {n, 0, 100}] (* _Emanuele Munarini_, Sep 01 2017 *)
%o A202826 (PARI) {a(n)=n!*polcoeff(exp(1/(1-x +x*O(x^n))^3-1),n)}
%o A202826 (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o A202826 {Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
%o A202826 {a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k) *(-1)^(n-k)*3^k)}
%o A202826 (Maxima)
%o A202826 makelist(sum(abs(stirling1(n,k))*3^k*belln(k),k,0,n),n,0,12); /* _Emanuele Munarini_, Sep 01 2017 */
%Y A202826 Cf. A000262, A136658, A202824.
%K A202826 nonn
%O A202826 0,2
%A A202826 _Paul D. Hanna_, Dec 25 2011
%E A202826 Example corrected by _Vaclav Kotesovec_, Feb 12 2013