OFFSET
0,3
COMMENTS
The maximal coefficient is that of x^[3n/2]. - M. F. Hasler, Jul 23 2007
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n=0..200
V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.
T. Neuschel, A Note on Extended Binomial Coefficients, J. Int. Seq. 17 (2014) # 14.10.4.
Claudia Smith and Verner E. Hoggatt, Jr. , A Study of the Maximal Values in Pascal's Quadrinomial Triangle, Fibonacci Quart. 17 (1979), no. 3, 264-269.
FORMULA
Limit_{n -> infinity} a(n+1)/a(n) = 4; for n>2, a(n+1) < 4*a(n). - Benoit Cloitre, Sep 28 2002
a(n) ~ 4^n * sqrt(2/(5*Pi*n)). - Vaclav Kotesovec, Aug 09 2013
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
MATHEMATICA
With[{exp=Total[x^Range[0, 3]]}, Table[Max[CoefficientList[Expand[exp^n], x]], {n, 0, 30}]] (* Harvey P. Dale, Nov 24 2011 *)
PROG
(PARI) a(n)=vecmax(vector(3*n, i, polcoeff((1+x+x^2+x^3)^n, i, x)))
(PARI) A005190(n)=polcoeff((1+x+x^2+x^3)^n, (3*n)>>1) \\ M. F. Hasler, Jul 23 2007
(Magma) P<x>:=PolynomialRing(Integers()); [Max(Coefficients((1+x+x^2+x^3)^n)): n in [0..26]]; // Vincenzo Librandi, Aug 09 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved