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%I M3112 N1261 #51 Sep 08 2022 08:44:29
%S 1,3,25,299,4785,95699,2296777,64309755,2057912161,74084837795,
%T 2963393511801,130389314519243,6258687096923665,325451729040030579,
%U 18225296826241712425,1093517809574502745499,69985139812768175711937,4758989507268235948411715
%N Expansion of e.g.f. exp(-x)/(1-4*x).
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 83.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Harvey P. Dale, <a href="/A001907/b001907.txt">Table of n, a(n) for n = 0..350</a>
%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%F a(n) = Sum_{k=0..n} (-1)^(n+k)*C(n,k)*k!*4^k. - _Ralf Stephan_, May 22 2004
%F Recurrence: a(n) = (4*n-1)*a(n-1) + 4*(n-1)*a(n-2). - _Vaclav Kotesovec_, Aug 16 2013
%F a(n) ~ n! * exp(-1/4)*4^n. - _Vaclav Kotesovec_, Aug 16 2013
%F E.g.f. A(x) = exp(-x)/(1-4x) satisfies (1-4x)A' - (3+4x)A = 0. - _Gheorghe Coserea_, Aug 06 2015
%F a(n) = exp(-1/4)*4^n*Gamma(n+1,-1/4), where Gamma is the incomplete Gamma function. - _Robert Israel_, Aug 07 2015
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (4*k - 1) * a(n-k). - _Ilya Gutkovskiy_, Jan 17 2020
%p f:= gfun:-rectoproc({a(n) = (4*n-1)*a(n-1) + 4*(n-1)*a(n-2), a(0)=1, a(1)=3},a(n), remember):
%p map(f, [$0..30]); # _Robert Israel_, Aug 07 2015
%t With[{nn=20},CoefficientList[Series[Exp[-x]/(1-4x),{x,0,nn}], x] Range[0,nn]!] (* or *) Table[Sum[(-1)^(n+k) Binomial[n,k]k! 4^k, {k,0,n}], {n,0,20}](* _Harvey P. Dale_, Oct 25 2011 *)
%t Join[{1}, RecurrenceTable[{a[1] == 3, a[2] == 25, a[n] == (4 n - 1) a[n-1] + 4(n - 1) a[n-2]}, a, {n, 20}]] (* _Vincenzo Librandi_, Aug 08 2015 *)
%o (PARI) a(n)=sum(k=0,n,(-1)^(n+k)*binomial(n,k)*k!*4^k)
%o (PARI) x = 'x+O('x^33); Vec(serlaplace(exp(-x)/(1-4*x))) \\ _Gheorghe Coserea_, Aug 06 2015
%o (Magma) I:=[3,25]; [1] cat [n le 2 select I[n] else (4*n-1)*Self(n-1)+4*(n-1)*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Aug 08 2015
%Y Cf. A000166, A000354, A000180, A001908.
%Y Column k=4 of A320032.
%K easy,nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Ralf Stephan_, May 22 2004
%E Typo fixed by _Charles R Greathouse IV_, Oct 28 2009
%E Name clarified by _Ilya Gutkovskiy_, Jan 17 2020
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