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

1, 4, 20, 116, 756, 5428, 42356, 355636, 3188340, 30333492, 304716148, 3218555700, 35618229364, 411717043252, 4957730174836, 62045057731892, 805323357485684, 10820999695801908, 150271018666120564, 2153476417340487476

Offset: 0

Author

Paul D. Hanna, Dec 18 2002

Keywords

• nonn

Comments

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

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

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

Exponential self-convolution of A001861. - Vladimir Reshetnikov, Oct 06 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Frank Simon, Algebraic Methods for Computing the Reliability of Networks, 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

Crossrefs

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.

Programs

Maple

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
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0, 4^m,
      add(b(n-1, max(m, j)), j=1..m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 03 2021

Mathematica

Table[n!, {n, 0, 20}]CoefficientList[Series[E^(4E^x-4), {x, 0, 20}], x]
Table[BellB[n,4],{n,0,20}] (* Vaclav Kotesovec, Mar 12 2014 *)
With[{nn=20},CoefficientList[Series[Exp[4(Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 03 2022 *)

Sage

expnums(20, 4) # Zerinvary Lajos, Jun 26 2008

Formula

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

E.g.f.: exp(4*(exp(x)-1)).

a(n) = exp(-4)*Sum_{k>=0} 4^k*k^n/k!. - Benoit Cloitre, Sep 25 2003

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

a(n) = Sum_{k = 0..n} 4^k*A048993(n, k); A048993: Stirling2 numbers. - Philippe Deléham, May 09 2004

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

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

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

G.f.: Sum_{j>=0} 4^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 07 2019

Extensions

More terms from R. J. Mathar, May 30 2008

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar