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A005721 Central quadrinomial coefficients.
(Formerly M3681)
5
1, 4, 44, 580, 8092, 116304, 1703636, 25288120, 379061020, 5724954544, 86981744944, 1327977811076, 20356299454276, 313095240079600, 4829571309488760, 74683398325804080, 1157402982351003420, 17971185794898859248 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Sum of squares of entries in the n-th row of triangle of quadrinomial coefficients (Pascal triangle of order 4). [Adi Dani, Jul 03 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 and Robert Israel, Table of n, a(n) for n = 0..597(n=0..100 from T. D. Noe)

Adi Dani, Restricted compositions of natural numbers-section: Generalized Pascal triangle

FORMULA

a(n) = A005190(2*n) = A008287(2*n, 3*n).

G.f.:  Let Z(x) be a solution of (-1+16*x)*(32*x-27)^2*Z^6+9*(-9+64*x)*(32*x-27)*Z^4+81*(80*x-27)*Z^2+729 = 0, with Z(0)=1. Compute a Puiseux series for Z(x) at x=0, then Z(x) in C[[x^(1/3)]].  Remove all non-integer powers of x.  The result is the generating function for A005721.  - Mark van Hoeij, Oct 29 2011

G.f.: F(G^(-1)(x)) where F(t) = (t^2-1)*(6*t+t^2+1)^(1/2)/(3*t^3+13*t^2+t-1) and G(t) = t/((t+1)^2*(6*t+t^2+1)). - Mark van Hoeij, Oct 30 2011

MAPLE

F := (t^2-1)*(6*t+t^2+1)^(1/2)/(3*t^3+13*t^2+t-1); G := t/((t+1)^2*(6*t+t^2+1));

Ginv := RootOf(numer(G-x), t); series(eval(F, t=Ginv), x=0, 20);

seq(coeff((1+x+x^2+x^3)^(2*n), x, 3*n), n=0..50); # Robert Israel, Nov 01 2015

MATHEMATICA

Table[Sum[(-1)^k*Binomial[2*n, k]*Binomial[5*n-4*k-1, 3*n-4*k], {k, 0, 3*n/4}], {n, 0, 25}] (* Adi Dani, Jul 03 2011 *)

PROG

(PARI) a(n)={local(v=Vec((1+x+x^2+x^3)^n)); sum(k=1, #v, v[k]^2); }

(PARI) a(n)=sum(k=0, 3*n/4, (-1)^k*binomial(2*n, k)*binomial(5*n-4*k-1, 3*n-4*k));

(PARI) vector(30, n, n--; polcoeff((1+x+x^2+x^3)^(2*n), (6*n)>>1)) \\ Altug Alkan, Nov 01 2015

CROSSREFS

Sequence in context: A223053 A222288 A053315 * A103870 A056063 A218224

Adjacent sequences:  A005718 A005719 A005720 * A005722 A005723 A005724

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified July 25 15:34 EDT 2017. Contains 289795 sequences.