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Central quadrinomial coefficients.
(Formerly M3681)
11

%I M3681 #106 Oct 28 2023 11:42:51

%S 1,4,44,580,8092,116304,1703636,25288120,379061020,5724954544,

%T 86981744944,1327977811076,20356299454276,313095240079600,

%U 4829571309488760,74683398325804080,1157402982351003420,17971185794898859248

%N Central quadrinomial coefficients.

%C Sum of squares of entries in the n-th row of triangle of quadrinomial coefficients A008287 (Pascal triangle of order 4). - _Adi Dani_, Jul 03 2011

%C Central coefficients in triangle A008287 ((1 + x + x^2 + x^3)^n), see link. - _Zagros Lalo_, Sep 25 2018

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.

%D Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 601, 602.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Robert Israel, <a href="/A005721/b005721.txt">Table of n, a(n) for n = 0..597</a> (first 101 terms from T. D. Noe)

%H Hacène Belbachir and Yassine Otmani, <a href="http://math.colgate.edu/~integers/x27/x27.pdf">Quadrinomial-Like Versions for Wolstenholme, Morley and Glaisher Congruences</a>, Integers (2023) Vol. 23.

%H Adi Dani, <a href="https://oeis.org/wiki/User:Adi_Dani_/Restricted_compositions_of_natural_numbers">Restricted compositions of natural numbers-section: Generalized Pascal triangle</a>

%H R. K. Guy, editor, <a href="/A339579/a339579.pdf">Western Number Theory Problems, 1985-12-21 & 23</a>, Typescript, Jul 13 1986, Dept. of Math. and Stat., Univ. Calgary, 11 pages. Annotated scan of pages 1, 3, 7, 9, with permission. See Problem 85:02.

%H R. K. Guy, <a href="/A005712/a005712.pdf">Letter to N. J. A. Sloane, 1987</a>

%H Zagros Lalo, <a href="/A005721/a005721_2.pdf">Formula for the Central coefficients in triangle A008287 ((1 + x + x^2 + x^3)^n).</a>

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

%F 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

%F 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

%F From _Bradley Klee_, Jun 25 2018: (Start)

%F 128*(n-1)*(2*n-3)*(2*n-1)*(5*n-1)*a(n-2) - 8*(2*n-1)*(145*n^3-319*n^2+201*n-30)*a(n-1) + 3*n*(3*n-2)*(3*n-1)*(5*n-6)*a(n) = 0.

%F G.f. G(x) satisfies a Picard-Fuchs type differential equation, 0 = Sum_{m=0..5, n=0..3} M_{m,n} x^m*(d^n/dx^n G(x)), with integer matrix:

%F M={{ 24, -6, 0, 0},

%F {-768, 1488, -54, 0},

%F {6144, -16128, 2520, -27},

%F { 0, 55296, -29568, 896},

%F { 0, 0, 49152, -7936},

%F { 0, 0, 0, 8192}}(End)

%F a(n) = sum_{k=0..floor(3n/4)} (-1)^k binomial(2n,k) * binomial(5n-4k-1,3n-4k). - _Muniru A Asiru_, Sep 26 2018

%F a(n) = Sum_{i=0..n} Sum_{j=n..2n}(f); f= ( (2*n)!/((j - n)!*(3*n + i - 2*j)!*(j - 2*i)!*i!) ); f=0 for (3*n + i - 2*j)<0 or (j - 2*i)<0. See also formula in Links section. - _Zagros Lalo_, Sep 27 2018

%p 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));

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

%p seq(coeff((1+x+x^2+x^3)^(2*n),x,3*n),n=0..50); # _Robert Israel_, Nov 01 2015

%t 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 *)

%t RecurrenceTable[{128*(n-1)*(2*n-3)*(2*n-1)*(5*n-1)*a[n-2] -8*(2*n-1)*(145*n^3-319*n^2+201*n-30)*a[n-1]+3*n*(3*n-2)*(3*n-1)*(5*n-6)*a[n]==0,

%t a[0]==1,a[1]==4},a,{n,0,5000}] (* _Bradley Klee_, Jun 25 2018 *)

%t a[n_] := a[n] = Sum[(2*n)!/((j - n)!*(3*n + i - 2*j)!*(j - 2*i)!*i!), {i, 0, n}, {j, n, 2*n}]; Table[a[n], {n, 0, 20}] (* _Zagros Lalo_, Sep 25 2018 *)

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

%o (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));

%o (PARI) vector(30, n, n--; polcoeff((1+x+x^2+x^3)^(2*n), (6*n)>>1)) \\ _Altug Alkan_, Nov 01 2015

%o (GAP) List([0..20],n->Sum([0..Int(3*n/4)],k->(-1)^k*Binomial(2*n,k)*Binomial(5*n-4*k-1,3*n-4*k))); # _Muniru A Asiru_, Sep 26 2018

%o (Magma) [(&+[(-1)^k*Binomial(2*n,k)*Binomial(5*n-4*k-1,3*n-4*k): k in [0..Floor(3*n/4)]]): n in [0..30]]; // _G. C. Greubel_, Oct 06 2018

%Y Cf. A002426, A005190, A008287.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_