A202824 - OEIS

%I #22 Sep 08 2022 08:46:01

%S 1,4,28,232,2248,24544,295456,3869632,54555328,821239552,13115934976,

%T 221076780544,3915685846528,72609585620992,1405168845395968,

%U 28302270409560064,591874919018500096,12824294700196052992,287350628454224478208,6647086535396002004992

%N Expansion of e.g.f.: exp( (1+x)^4 - 1 ).

%H Seiichi Manyama, <a href="/A202824/b202824.txt">Table of n, a(n) for n = 0..501</a>

%F a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(k) * 4^k.

%F a(n) ~ n^(3*n/4)*2^(n/2-1)*exp(-3/4+5/16*sqrt(2)*n^(1/4)+sqrt(2)*n^(3/4)-3/4*n+3/4*sqrt(n)). - _Vaclav Kotesovec_, May 23 2013

%F a(n+4) - 4*a(n+3) - 12*(n+3)*a(n+2) - 12*(n+2)*(n+3)*a(n+1) - 4*(n+1)*(n+2)*(n+3)*a(n) = 0. - _Emanuele Munarini_, Aug 31 2017

%e E.g.f.: A(x) = 1 + 4*x + 28*x^2/2! + 232*x^3/3! + 2248*x^4/4! +...

%e where A(x) = exp(4*x + 6*x^2 + 4*x^3 + x^4).

%t CoefficientList[Series[Exp[(1+x)^4-1], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, May 23 2013 *)

%t Table[Sum[ (-1)^(n - k) Abs[StirlingS1[n, k]] 4^k BellB[k], {k, 0, n}], {n, 0, 100}] (* _Emanuele Munarini_, Aug 31 2017 *)

%o (PARI) {a(n)=n!*polcoeff(exp((1+x +x*O(x^n))^4-1),n)}

%o (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

%o {Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}

%o {a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k) * 4^k)}

%o (Maxima)

%o a(n) := sum((-1)^(n-k)*abs(stirling1(n,k))*4^k*belln(k),k,0,n);

%o makelist(a(n),n,0,12); /* _Emanuele Munarini_, Aug 31 2017 */

%o (Magma) [(&+[4^k*Bell(k)*StirlingFirst(n,k): k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Jul 25 2019

%o (Sage) [sum((-1)^(n-k)*4^k*bell_number(k)*stirling_number1(n,k) for k in (0..n)) for n in (0..20)] # _G. C. Greubel_, Jul 25 2019

%o (GAP) List([0..20], n-> Sum([0..n], k-> (-1)^(n-k)*4^k*Bell(k)* Stirling1(n,k) )); # _G. C. Greubel_, Jul 25 2019

%Y Cf. A000110, A000898, A008275, A192989, A202825.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 25 2011