A005190 Central quadrinomial coefficients: largest coefficient of (1 + x + x^2 + x^3)^n.
1, 1, 4, 12, 44, 155, 580, 2128, 8092, 30276, 116304, 440484, 1703636, 6506786, 25288120, 97181760, 379061020, 1463609356, 5724954544, 22187304112, 86981744944, 338118529539, 1327977811076, 5175023913008, 20356299454276
Offset: 0
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.
- Index entries for sequences of k-nomial coefficients
Programs
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Magma
P
:=PolynomialRing(Integers()); [Max(Coefficients((1+x+x^2+x^3)^n)): n in [0..26]]; // Vincenzo Librandi, Aug 09 2014 -
Mathematica
With[{exp=Total[x^Range[0,3]]},Table[Max[CoefficientList[Expand[exp^n], x]],{n,0,30}]] (* Harvey P. Dale, Nov 24 2011 *) -
PARI
a(n)=vecmax(vector(3*n,i,polcoeff((1+x+x^2+x^3)^n,i,x)))
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PARI
A005190(n)=polcoeff((1+x+x^2+x^3)^n,(3*n)>>1) \\ M. F. Hasler, Jul 23 2007
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
Comments