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A104631
Coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.
2
0, 1, 4, 18, 80, 365, 1686, 7875, 37080, 175725, 837100, 4004770, 19227924, 92599533, 447118140, 2163837030, 10492874384, 50972030189, 248000853348, 1208335275170, 5894873067200, 28791371852145, 140768761906190
OFFSET
0,3
COMMENTS
In the triangle of pentanomial coefficients, these numbers are in the column next to the central pentanomial coefficients, A005191. Note that for n>0, n divides a(n). This divisibility property is also true for the triangle of trinomial coefficients, A027907, but apparently for no other triangle of m-nomial coefficients. The quotient a(n)/n is in A104632.
LINKS
FORMULA
G.f.: sqrt((5*x^2+2*x-1+(x+1)*sqrt(5*x^2-6*x+1))/(2*x*(1-x)*(5*x+4)*(5*x-1))). - Mark van Hoeij, Nov 16 2011
Recurrence: 2*(n-1)*(2*n+1)*a(n) = (19*n^2 - 19*n + 2)*a(n-1) + 5*(2*n^2 - 3*n - 1)*a(n-2) - 25*(n-2)*n*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 5^n/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 17 2012
a(n) = n * A104632(n) for n>=0. - Michael Somos, Aug 12 2018
EXAMPLE
G.f. = x + 4*x^2 + 18*x^3 + 80*x^4 + 365*x^5 + 1686*x^6 + 7875*x^7 + ... - Michael Somos, Aug 12 2018
MATHEMATICA
f=1; Table[f=Expand[f(x^4+x^3+x^2+x+1)]; Coefficient[f, x, 2n+1], {n, 30}]
PROG
(Magma) P<x>:=PolynomialRing(Integers()); [n eq 0 select 0 else Coefficients((1+x+x^2+x^3+x^4)^n)[2*n+2]: n in [0..22]]; // Bruno Berselli, Nov 17 2011
(PARI) x='x+O('x^30); concat([0], Vec(sqrt((5*x^2+2*x-1+(x+1)*sqrt(5*x^2-6*x+1))/(2*x*(1-x)*(5*x+4)*(5*x-1))))) \\ G. C. Greubel, Aug 12 2018
CROSSREFS
Cf. A035343 (triangle of pentanomial coefficients).
Cf. A104632.
Sequence in context: A045902 A090017 A257390 * A106391 A063881 A264004
KEYWORD
nonn,easy
AUTHOR
T. D. Noe, Mar 17 2005
STATUS
approved