A235391 Duplicate of A129775.
1, 1, 2, 6, 21, 78, 298, 1157, 4539, 17936, 71251, 284188, 1137076, 4561093, 18333337, 73816489, 297635750, 1201551286, 4855672249, 19640147061, 79501958895, 322037615290, 1305256267511, 5293166568270, 21475362822956, 87166344495561, 353933533606927
Offset: 0
Keywords
Examples
G.f. = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 78*x^5 + 298*x^6 + 1157*x^7 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
-
Magma
m:=25; R
:=PowerSeriesRing(Rationals(), m); Coefficients(R!(2/( 2-x - x/Sqrt(1-4*x)))); // G. C. Greubel, Aug 07 2018 -
Mathematica
a[ n_] := SeriesCoefficient[ 2 / (2 - x - x / Sqrt[1 - 4 x]), {x, 0, n}] -
PARI
{a(n) = if( n<0, 0, polcoeff( 2 / (2 - x - x / sqrt(1 - 4*x + x * O(x^n))), n))}
Formula
G.f.: 1 / (1 - x / (1 - x / (1 - 2*x / (1 - x / (2 - 3*x / (1 - 2*x / (3 - 4*x / ... ))))))).
D-finite with recurrence: 0 = (4*n + 6) * a(n) - (17*n + 27) * a(n+1) + (24*n + 42) * a(n+2) - (9*n + 21) * a(n+3) + (n + 3) * a(n+4). - Sign flipped by R. J. Mathar, Feb 16 2020
0 = a(n) * (16*a(n+1) - 74*a(n+2) + 120*a(n+3) - 66*a(n+4) + 10*a(n+5))+ a(n+1) * (-62*a(n+1) + 361*a(n+2) - 480*a(n+3) + 265*a(n+4) - 41*a(n+5)) + a(n+2) * (-342*a(n+2) + 615*a(n+3) - 335*a(n+4) + 54*a(n+5)) + a(n+3) * (-90*a(n+3) + 75*a(n+4) - 15*a(n+5)) + a(n+4) * (-3*a(n+4) + a(n+5)).
a(n) = A129775(n) if n>0.
HANKEL transform is A000012.
INVERT transform is A073525.