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

0,2

Links

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

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)

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 *)

Prog

(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
(SageMath)
def A144647_list(prec):
    P.<x> = 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

Crossrefs

Cf. A001515, A144301, A144498.

Sequence in context: A337167 A140094 A284859 * A215505 A182304 A060910

Adjacent sequences: A144644 A144645 A144646 * A144648 A144649 A144650

Keyword

nonn

Author

N. J. A. Sloane, Jan 26 2009

Extensions

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

Status

approved