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%I M3456 #41 Sep 08 2022 08:44:33
%S 1,1,4,12,44,155,580,2128,8092,30276,116304,440484,1703636,6506786,
%T 25288120,97181760,379061020,1463609356,5724954544,22187304112,
%U 86981744944,338118529539,1327977811076,5175023913008,20356299454276
%N Central quadrinomial coefficients: largest coefficient of (1 + x + x^2 + x^3)^n.
%C The maximal coefficient is that of x^[3n/2]. - _M. F. Hasler_, Jul 23 2007
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A005190/b005190.txt">Table of n, a(n) for n=0..200</a>
%H V. E. Hoggatt, Jr. and M. Bicknell, <a href="http://www.fq.math.ca/Scanned/7-4/hoggatt-a.pdf">Diagonal sums of generalized Pascal triangles</a>, Fib. Quart., 7 (1969), 341-358, 393.
%H T. Neuschel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Neuschel/neuschel4.html">A Note on Extended Binomial Coefficients</a>, J. Int. Seq. 17 (2014) # 14.10.4.
%H Claudia Smith and Verner E. Hoggatt, Jr. , <a href="http://www.fq.math.ca/Scanned/17-3/smith.pdf">A Study of the Maximal Values in Pascal's Quadrinomial Triangle</a>, Fibonacci Quart. 17 (1979), no. 3, 264-269.
%F Limit_{n -> infinity} a(n+1)/a(n) = 4; for n>2, a(n+1) < 4*a(n). - _Benoit Cloitre_, Sep 28 2002
%F a(n) ~ 4^n * sqrt(2/(5*Pi*n)). - _Vaclav Kotesovec_, Aug 09 2013
%F Recurrence: 3*n*(3*n-1)*(3*n+1)*(75*n^3 - 390*n^2 + 635*n - 348)*a(n) = 12*(675*n^5 - 4095*n^4 + 8405*n^3 - 7925*n^2 + 3548*n - 664)*a(n-1) + 16*(n-1)*(2175*n^5 - 13335*n^4 + 29275*n^3 - 27707*n^2 + 11334*n - 2814)*a(n-2) - 640*(n-2)*(n-1)*(15*n^3 - 66*n^2 + 52*n - 15)*a(n-3) - 512*(n-3)*(n-2)*(n-1)*(75*n^3 - 165*n^2 + 80*n - 28)*a(n-4). - _Vaclav Kotesovec_, Aug 09 2013
%t With[{exp=Total[x^Range[0,3]]},Table[Max[CoefficientList[Expand[exp^n], x]],{n,0,30}]] (* _Harvey P. Dale_, Nov 24 2011 *)
%o (PARI) a(n)=vecmax(vector(3*n,i,polcoeff((1+x+x^2+x^3)^n,i,x)))
%o (PARI) A005190(n)=polcoeff((1+x+x^2+x^3)^n,(3*n)>>1) \\ _M. F. Hasler_, Jul 23 2007
%o (Magma) P<x>:=PolynomialRing(Integers()); [Max(Coefficients((1+x+x^2+x^3)^n)): n in [0..26]]; // _Vincenzo Librandi_, Aug 09 2014
%Y Cf. A001405, A002426, A005191, A018901, A025012, A025013, A025014.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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