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A005725 Quadrinomial coefficients.
(Formerly M2843)
11
1, 1, 3, 10, 31, 101, 336, 1128, 3823, 13051, 44803, 154518, 534964, 1858156, 6472168, 22597760, 79067375, 277164295, 973184313, 3422117190, 12049586631, 42478745781, 149915252028, 529606271560, 1872653175556, 6627147599476, 23471065878276, 83186110269928 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Coefficient of x^n in (1+x+x^2+x^3)^n.

Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2), (1,3). - Joerg Arndt, Jul 05 2011

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

R. K. Guy, Letter to N. J. A. Sloane, 1987

FORMULA

a(n) = sum(i+j+k=n, 0<=k<=j<=i<=n, C(n,i)*C(i,j)*C(j,k)). - Benoit Cloitre, Jun 06 2004

G.f.: A(x) where (16*x^3+8*x^2+11*x-4)*A(x)^3+(3-2*x)*A(x)+1 = 0. - Mark van Hoeij, Apr 30 2013

Recurrence: 2*n*(2*n-1)*(13*n-19)*a(n) = (143*n^3 - 352*n^2 + 251*n - 54)*a(n-1) + 4*(n-1)*(26*n^2 - 51*n + 15)*a(n-2) + 16*(n-2)*(n-1)*(13*n-6)*a(n-3). - Vaclav Kotesovec, Aug 10 2013

a(n) ~ sqrt((39+7*39^(2/3)/c+39^(1/3)*c)/156) * ((b+11+217/b)/12)^n/sqrt(Pi*n), where b = (6371+624*sqrt(78))^(1/3), c = (117+2*sqrt(78))^(1/3). - Vaclav Kotesovec, Aug 10 2013

a(n) = A008287(n, n). - Sean A. Irvine, Aug 15 2016

a(n) = hypergeom([1/2-n/2, -n, -n/2], [1/2, 1], -1). - Vladimir Reshetnikov, Oct 04 2016

EXAMPLE

For n=2, (x^3 + x^2 + x + 1)^2 = x^6 + 2x^5 + 3x^4 + 4x^3 + 3x^2 + 2x + 1, and the coefficient of x^n = x^2 is 3, so a(2) = 3. - Michael B. Porter, Aug 15 2016

MAPLE

seq(add(binomial(n, 2*k)*binomial(n, k), k=0..floor(n/2)), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001

a := n -> add(binomial(n, j)*binomial(n, 2*j), j=0..n): seq(a(n), n=1..25); # Zerinvary Lajos, Feb 12 2007

seq(coeff(series(RootOf((16*x^3+8*x^2+11*x-4)*A^3+(3-2*x)*A+1, A), x=0, n+1), x, n), n=0..30);  # Mark van Hoeij, Apr 30 2013

MATHEMATICA

a[n_] := Coefficient[(1+x+x^2+x^3)^n, x^n]; a[0] = 1; Table[a[n], {n, 0, 25}] (* Jean-Fran├žois Alcover, Nov 15 2011 *)

Table[HypergeometricPFQ[{1/2 - n/2, -n, -n/2}, {1/2, 1}, -1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 04 2016 *)

PROG

(Maxima) quadrinomial(n, k):=coeff(expand((1+x+x^2+x^3)^n), x, k); makelist(quadrinomial(n, n), n, 0, 12); \\ Emanuele Munarini, Mar 15 2011

(MAGMA) P<x>:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2+x^3)^n)[n+1]: n in [0..25] ]; // Bruno Berselli, Jul 05 2011

(PARI) a(n)=my(x='x); polcoeff((x^3+x^2+x+1)^n, n) \\ Charles R Greathouse IV, Feb 07 2017

CROSSREFS

Cf. A008287.

Column k=3 of A305161.

Sequence in context: A114487 A017934 A005510 * A302287 A079522 A024426

Adjacent sequences:  A005722 A005723 A005724 * A005726 A005727 A005728

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, Jul 12 2000

STATUS

approved

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Last modified November 16 23:51 EST 2018. Contains 317275 sequences. (Running on oeis4.)