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A078944 First column of A078939, the fourth power of lower triangular matrix A056857.

%I A078944 #64 Jan 11 2025 18:43:50
%S A078944 1,4,20,116,756,5428,42356,355636,3188340,30333492,304716148,
%T A078944 3218555700,35618229364,411717043252,4957730174836,62045057731892,
%U A078944 805323357485684,10820999695801908,150271018666120564,2153476417340487476
%N A078944 First column of A078939, the fourth power of lower triangular matrix A056857.
%C A078944 Also, the number of ways of placing n labeled balls into n unlabeled (but 4-colored) boxes. Binomial transform of this sequence is A078945 and a(n+1) = 4*A078945(n). - _Paul D. Hanna_, Dec 08 2003
%C A078944 First column of PE^4, where PE is given in A011971, second power in A078937, third power in A078938, fourth power in A078939. - _Gottfried Helms_, Apr 08 2007
%C A078944 The number of ways of putting n labeled balls into a set of bags and then putting the bags into 4 labeled boxes. - _Peter Bala_, Mar 23 2013
%C A078944 Exponential self-convolution of A001861. - _Vladimir Reshetnikov_, Oct 06 2016
%H A078944 Vincenzo Librandi, <a href="/A078944/b078944.txt">Table of n, a(n) for n = 0..200</a>
%H A078944 Frank Simon, <a href="https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa-101154">Algebraic Methods for Computing the Reliability of Networks</a>, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012. See Table 5.1. - From _N. J. A. Sloane_, Jan 04 2013
%F A078944 PE=exp(matpascal(5))/exp(1); A = PE^4; a(n)= A[ n,1 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^4; a(n)=A[ n,1]. - _Gottfried Helms_, Apr 08 2007
%F A078944 E.g.f.: exp(4*(exp(x)-1)).
%F A078944 a(n) = exp(-4)*Sum_{k>=0} 4^k*k^n/k!. - _Benoit Cloitre_, Sep 25 2003
%F A078944 G.f.: 4*(x/(1-x))*A(x/(1-x)) = A(x) - 1; four times the binomial transform equals this sequence shifted one place left. - _Paul D. Hanna_, Dec 08 2003
%F A078944 a(n) = Sum_{k = 0..n} 4^k*A048993(n, k); A048993: Stirling2 numbers. - _Philippe Deléham_, May 09 2004
%F A078944 G.f.: (G(0) - 1)/(x-1)/4 where G(k) = 1 - 4/(1-k*x)/(1-x/(x-1/G(k+1))); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 16 2013
%F A078944 G.f.: T(0)/(1-4*x), where T(k) = 1 - 4*x^2*(k+1)/(4*x^2*(k+1) - (1-(k+4)*x)*(1-(k+5)*x)/T(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Oct 28 2013
%F A078944 a(n) ~ n^n * exp(n/LambertW(n/4)-4-n) / (sqrt(1+LambertW(n/4)) * LambertW(n/4)^n). - _Vaclav Kotesovec_, Mar 12 2014
%F A078944 G.f.: Sum_{j>=0} 4^j*x^j / Product_{k=1..j} (1 - k*x). - _Ilya Gutkovskiy_, Apr 07 2019
%p A078944 A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A078939 := proc(n,c) add( A078938(n,k)*A056857(k+1,c),k=0..n) ; end: A078944 := proc(n) A078939(n+1,0) ; end: seq(A078944(n),n=0..25) ; # _R. J. Mathar_, May 30 2008
%p A078944 # second Maple program:
%p A078944 b:= proc(n, m) option remember; `if`(n=0, 4^m,
%p A078944       add(b(n-1, max(m, j)), j=1..m+1))
%p A078944     end:
%p A078944 a:= n-> b(n, 0):
%p A078944 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 03 2021
%t A078944 Table[n!, {n, 0, 20}]CoefficientList[Series[E^(4E^x-4), {x, 0, 20}], x]
%t A078944 Table[BellB[n,4],{n,0,20}] (* _Vaclav Kotesovec_, Mar 12 2014 *)
%t A078944 With[{nn=20},CoefficientList[Series[Exp[4(Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, May 03 2022 *)
%o A078944 (Sage) expnums(20, 4) # _Zerinvary Lajos_, Jun 26 2008
%Y A078944 Cf. A000110, A001861, A027710, A056857, A078937, A078938, A078939, A078944, A078945, A129323, A129324, A129325, A129327, A129328, A129329, A129331, A129332, A129333, A144180, A144223, A144263, A189233, A221159, A221176.
%K A078944 nonn
%O A078944 0,2
%A A078944 _Paul D. Hanna_, Dec 18 2002
%E A078944 More terms from _R. J. Mathar_, May 30 2008
%E A078944 Edited by _N. J. A. Sloane_, Jul 02 2008 at the suggestion of _R. J. Mathar_