A178694 -id:A178694 - cp's OEIS frontend

A178693 Numerators of coefficients of Maclaurin series for 2 - sqrt(1 - x - x^2).

1, 1, 5, 5, 45, 95, 465, 1165, 24445, 65595, 359915, 1003315, 11342185, 32415435, 187063145, 544172445, 25508284445, 75196195795, 445774614215, 1327748661015, 15887874844835, 47715177777185, 287618252461095, 869652752181595

Offset: 0

Clark Kimberling, Jun 04 2010

Every term after the second is a multiple of 5.

			The Maclaurin series begins with 1 + (1/2)*x + (5/8)*x^2 + (5/16)*x^3 + ....

M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 41.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A178694.

Cf. A046161 (denominators).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 2-Sqrt(1-x-x^2) )); [Numerator(Factorial(n-1)*b[n]): n in [1..m]]; // G. C. Greubel, Jan 25 2019

Numerator[CoefficientList[Series[2-Sqrt[1-x-x^2], {x, 0, 30}], x]] (* G. C. Greubel, Jan 25 2019 *)

my(x='x+O('x^30)); v=Vec( 2-sqrt(1-x-x^2) ); vector(#v, n, numerator(v[n])) \\ G. C. Greubel, Jan 25 2019

G.f.: 2 - sqrt(1 - x - x^2) for the fractions (not the numerators).

A179191 Expansion of o.g.f. (1/2)(-1 + 1/sqrt(1 - 4x - 4*x^2)).

Original entry on oeis.org

0, 1, 4, 16, 68, 296, 1312, 5888, 26672, 121696, 558464, 2574848, 11917952, 55345408, 257741824, 1203224576, 5629027072, 26383656448, 123868321792, 582414688256, 2742116907008, 12926036258816, 60998951747584, 288147689046016, 1362407763795968, 6447125560016896

Offset: 0

Clark Kimberling, Jul 01 2010

nonn

G.f. A(x) satisfies A(x)^2 + A(x) = (x^2 + x)/(1 - 4*x - 4*x^2). - Michael Somos, Jan 28 2019

			G.f. = x + 4*x^2 + 16*x^3 + 68*x^4 + 296*x^5 + 1312*x^6 + 5888*x^7 + ....

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv:1410.5747 [math.CO], 2014.

Cf. A178693, A178694, A179190.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (-1 + 1/Sqrt(1-4*x-4*x^2))/2 )); // G. C. Greubel, Jan 25 2019

CoefficientList[1/2 (-1 + (1-4x-4x^2)^(-1/2)) + O[x]^23, x] (* Jean-François Alcover, Jul 27 2018 *)

a(n):=sum(m*sum(sum(binomial(i-1,k-1)*binomial(i,n-i),i,k,n)*sum(binomial(j,2*j-m-k)*binomial(k,j),j,0,k)/k,k,m,n),m,1,n); /* Vladimir Kruchinin, Mar 11 2011 */
a(n):=sum(2^(n-k-1)*binomial(n,k)*binomial(n-k,k),k,0,n); /* Vladimir Kruchinin, Mar 12 2015 */

my(x='x+O('x^30)); concat([0], Vec((-1 +1/sqrt(1-4*x-4*x^2))/2)) \\ G. C. Greubel, Jan 25 2019

((-1 + 1/sqrt(1-4*x-4*x^2))/2).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 25 2019

G.f.: (1/2)*(-1 + 1/sqrt(1 - 4*x - 4*x^2)).
G.f.: A(x) = x*A001006(A000045(x)/x-1)/(1-x*A001006(A000045(x)/x-1)).
a(n) = Sum_{m=1..n} m*Sum_{k=m..n} (Sum_{i=k..n} binomial(i-1,k-1)*binomial(i,n-i))*(Sum_{j=0..k} binomial(j,2*j-m-k)*binomial(k,j))/k. - Vladimir Kruchinin, Mar 11 2011
a(n) = Sum_{k=0..n} 2^(n-k-1)*binomial(n,k)*binomial(n-k,k). - Vladimir Kruchinin, Mar 12 2015
From Vaclav Kotesovec, Jan 26 2019: (Start)
D-finite with recurrence: n*a(n) = 2*(2*n - 1)*a(n-1) + 4*(n-1)*a(n-2).
a(n) ~ 2^(n - 7/4) * (1 + sqrt(2))^(n + 1/2) / sqrt(Pi*n). (End)
0 = a(n)*(16*a(n+1) +24*a(n+2) -8*a(n+3)) + a(n+1)*(+8*a(n+1) +16*a(n+2) -6*a(n+3)) + a(n+2)*(-2*a(n+2) +a(n+3)) for all n in Z except n=-1. - Michael Somos, Jan 27 2019
a(n) = A006139(n)/2, n>0. - R. J. Mathar, Jan 24 2020

A179190 Coefficient [x^n] of the Maclaurin series for 2 - sqrt(1 - 4x - 4x^2).

Original entry on oeis.org

1, 2, 4, 8, 24, 80, 288, 1088, 4256, 17088, 70016, 291584, 1230592, 5251584, 22623232, 98248704, 429677056, 1890700288, 8364824576, 37186449408, 166030266368, 744180244480, 3347321831424, 15104525959168, 68357598756864

Offset: 0

Clark Kimberling, Jul 01 2010

nonn

			The Maclaurin series is 1 + 2*x + 4*x^2 + 8*x^3 + 24*x^4 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..151

Cf. A178693, A178694, A179191.

A179190 := proc(n) if n = 0 then 1; else add( doublefactorial(2*n-2*k-3) *2^(n-k) / k! / (n-2*k)!, k=0..floor(n/2)) ; end if; end proc: # R. J. Mathar, Jul 11 2011

Table[SeriesCoefficient[Series[2-Sqrt[1-4*t-4*t^2], {t,0,n}], n], {n, 0, 30}] (* G. C. Greubel, Jan 25 2019 *)

makelist(coeff(taylor(2-sqrt(1-4*x-4*x^2), x, 0, n), x, n), n, 0, 24); /* Bruno Berselli, Jul 04 2011 */

G.f.: 2 - sqrt(1 - 4*x - 4*x^2).
a(n) = 4*A071356(n-2), n >= 2. - R. J. Mathar, Jul 08 2010
a(n) = Sum_{k=0..floor(n/2)} (2*n - 2*k - 3)!! *2^(n-k)/(k!*(n-2k)!), n > 0. - R. J. Mathar, Jul 11 2011
a(n) ~ 2^(n - 1/4) * (1 + sqrt(2))^(n - 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 26 2019
D-finite with recurrence: n*a(n) +2*(-2*n+3)*a(n-1) +4*(-n+3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020

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A178693 Numerators of coefficients of Maclaurin series for 2 - sqrt(1 - x - x^2).

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A179191 Expansion of o.g.f. (1/2)*(-1 + 1/sqrt(1 - 4*x - 4*x^2)).

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A179190 Coefficient [x^n] of the Maclaurin series for 2 - sqrt(1 - 4*x - 4*x^2).

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A179191 Expansion of o.g.f. (1/2)(-1 + 1/sqrt(1 - 4x - 4*x^2)).

A179190 Coefficient [x^n] of the Maclaurin series for 2 - sqrt(1 - 4x - 4x^2).