A005190 Central quadrinomial coefficients: largest coefficient of (1 + x + x^2 + x^3)^n. (Formerly M3456)

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

0,3

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

Cf. A001405, A002426, A005191, A018901, A025012, A025013, A025014.

Sequence in context: A167402 A320643 A060897 * A149360 A149361 A149362

Adjacent sequences: A005187 A005188 A005189 * A005191 A005192 A005193

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved