A144647 Second differences of A001515 (or A144301).

1, 4, 25, 199, 1936, 22411, 301939, 4649800, 80654599, 1556992441, 33120019516, 769887934729, 19419368959225, 528311452144204, 15421347559288441, 480784227676809991, 15945180393017896024, 560549114426134288675

Offset: 0

Author

N. J. A. Sloane, Jan 26 2009

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A001515, A144301, A144498.

Programs

Magma

[n le 2 select 4^(n-1) else ( ((2*n-3)*(4*n^2-12*n+13))*Self(n-1) + (4*n^2-8*n+7)*Self(n-2) )/(4*n^2-16*n+19): n in [1..30]]; // G. C. Greubel, Sep 28 2023

Maple

A001515 := proc(n) simplify(hypergeom([n+1,-n],[],-1/2)) ; end: A144647 := proc(n) if n =0 then A001515(n) ; else A001515(n+1)-2*A001515(n)+A001515(n-1) ; fi; end: seq(A144647(n),n=0..30) ; # R. J. Mathar, Feb 01 2009

Mathematica

Join[{1},Differences[RecurrenceTable[{a[0]==1,a[1]==2,a[n]== (2n-1)a[n-1]+ a[n-2]},a[n],{n,25}],2]] (* Harvey P. Dale, Jun 18 2011 *)

SageMath

def A144647_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (-1+4*x+2*(1-x)*sqrt(1-2*x))*exp(1-sqrt(1-2*x))/(sqrt(1-2*x))^3 ).egf_to_ogf().list()
A144647_list(40) # G. C. Greubel, Sep 28 2023

Formula

G.f.: (1-x)^2/(x*Q(0)) + 1 - 1/x, where Q(k)= 1 - x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013

G.f.: T(0)*(1-x)/x + 1 - 1/x, where T(k) = 1 - x*(k+1)/( x*(k+1) - (1-x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2013

From G. C. Greubel, Sep 28 2023: (Start)

a(n) = A001515(n+1) - 2*A001515(n) + A001515(n).

a(n) = 2*A001515(n+1) - (2*n+3)*A001515(n).

a(n) = ( ((2*n-1)*(4*n^2 - 4*n + 5))*a(n-1) + (4*n^2 + 3)*a(n-2) )/(4*n^2 - 8*n + 7), with a(0) = 1, a(1) = 4.

E.g.f.: (-1 + 4*x + 2*(1-x)*sqrt(1-2*x))*exp(1-sqrt(1-2*x))/(sqrt(1-2*x))^3. (End)

Extensions

More terms from R. J. Mathar, Feb 01 2009