A247195 Expansion of -(sqrt(x^4-4*x^3-6*x^2-4*x+1) +x^2-2*x-1)/4.

0, 1, 1, 3, 9, 33, 129, 531, 2265, 9921, 44361, 201651, 929073, 4328865, 20362137, 96562659, 461169873, 2216134401, 10707788721, 51988771107, 253515373305, 1241069449377, 6097106216529, 30050252046195, 148541591990505, 736237012296897

Offset

0,4

Links

Robert Israel, Table of n, a(n) for n = 0..1391

Formula

a(n) = Sum_{k=1..n-1} ( Sum_{i=0..n-k-1} 2^i*binomial(k,n-k-i-1)*binomial(k+i-1,k-1) )*binomial(n-k-1,k-1)/k, n>1, a(0)=0, a(1)=1.

D-finite with recurrence (n-2)*a(n)+(2-4*n)*a(n+1)+(-6-6*n)*a(n+2)+(-10-4*n)*a(n+3)+(n+4)*a(n+4)=0. - Robert Israel, Nov 26 2018

Maple

f:= gfun:-rectoproc({(n-2)*a(n)+(2-4*n)*a(n+1)+(-6-6*n)*a(n+2)+(-10-4*n)*a(n+3)+(n+4)*a(n+4)=0, a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 3, a(4) = 9}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Nov 26 2018

Mathematica

a[n_] := If[n == 1, 1, Sum[Sum[2^i*Binomial[k, n-k-i-1]*Binomial[k+i-1, k-1], {i, 0, n-k-1}]*Binomial[n-k-1, k-1]/k, {k, 1, n-1}]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 25 2014, translated from Maxima *)
CoefficientList[Series[-(Sqrt[x^4-4*x^3-6*x^2-4*x+1] + x^2-2*x-1)/4, {x, 0, 30}], x] (* G. C. Greubel, Nov 26 2018 *)

Prog

(Maxima)
a(n):=if n=1 then 1 else sum(((sum(2^i*binomial(k, n-k-i-1)*binomial(k+i-1, k-1), i, 0, n-k-1))*binomial(n-k-1, k-1))/k, k, 1, n-1);
(PARI) my(x='x+O('x^30)); concat([0], Vec(-(sqrt(x^4-4*x^3-6*x^2-4*x+1) +x^2-2*x-1)/4)) \\ G. C. Greubel, Nov 26 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!( -(Sqrt(x^4-4*x^3-6*x^2-4*x+1) + x^2-2*x-1)/4 )); // G. C. Greubel, Nov 26 2018
(Sage) s=(-(sqrt(x^4-4*x^3-6*x^2-4*x+1) +x^2-2*x-1)/4).series(x, 30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 26 2018

Crossrefs

Sequence in context: A084508 A151043 A151044 * A236408 A217617 A320181

Adjacent sequences: A247192 A247193 A247194 * A247196 A247197 A247198

Keyword

nonn

Author

Vladimir Kruchinin, Nov 24 2014

Status

approved