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A003148 a(n+1) = a(n) + 2n*(2n+1)*a(n-1), with a(0) = a(1) = 1.
(Formerly M4389)
13
1, 1, 7, 27, 321, 2265, 37575, 390915, 8281665, 114610545, 2946939975, 51083368875, 1542234996225, 32192256321225, 1114841223671175, 27254953356505875, 1064057291370698625, 29845288035840902625, 1296073464766972266375, 41049997128507054562875 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Numerators of sequence of fractions with e.g.f. 1/((1-x)*(1+x)^(1/2)). The denominators are successive powers of 2.

a(n) is the coefficient of x^n in arctan(sqrt(2*x/(1-x)))/sqrt(2*x*(1-x)) multiplied by (2*n+1)!!.

This sequence is the linking pin between the a(n) formulas of the ED1, ED2, ED3 and ED4 array rows, see A167552, A167565, A167580 and A167591. - Johannes W. Meijer, Nov 23 2009

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

P. S. Bruckman, An interesting sequence of numbers derived from various generating functions, Fib. Quart., 10 (1972), 169-181.

R. J. Mathar, Numerical Representation of the Incomplete Gamma Function of Complex Argument, arXiv:math/0306184 [math.NA], 2003-2004; cf. Eq. 22.

FORMULA

a(n) = (-1)^n*(2n-1)!! + 2n*a(n-1) with (2n-1)!! = 1*3*5*..*(2n-1) the double factorial. - R. J. Mathar, Jun 12 2003

a(n) = [(2n+1)!!/4] Int ([cos(phi)]^n cos(phi/2), phi=-Pi..Pi). - R. J. Mathar, Jun 30 2003

a(n) = (2n+1)!! 2F1(-n, 1/2;3/2;2). - R. J. Mathar, Jun 30 2003

In terms of the (terminating) Gauss hypergeometric function/series 2F1(., .; ., .) a(n) is a special case of the family of integer sequences defined by a(m, n) = [(2n+2m+1)!!/(2m+1)] 2F1(-n, m+1/2; m+3/2; 2), m=0, 1, 2, ..., n=0, 1, 2, ...; a(n) = a(0, n); a(m, n) = [(2n+2m+1)!!/4] Int ([sin(phi/2)]^(2m) [cos(phi)]^n cos(phi/2), phi=-Pi. .Pi); 4(n+1)a(m, n) = (2m-1) a(m-1, n+1)+(-1)^n (2n+2m+1)!!. a(0, n) = this sequence, a(1, n) = A077568, a(2, n) = A084543. - R. J. Mathar, Jun 30 2003

E.g.f.: 1/(sqrt(1+2*x)*(1-2*x)). - Vladeta Jovovic, Oct 12 2003

a(n) = (2^n)*n!*A123746(n)/A046161(n) = (2^n)*n!*sum(binomial(2*k,k)*(-1/4)^k,k=0..n). From the e.g.f. - Wolfdieter Lang, Oct 06 2008.

a(n) = A091520(n) * n! / 2^n. - Michael Somos, Mar 17 2011

EXAMPLE

arctan(sqrt(2*x/(1-x)))/sqrt(2*x*(1-x)) = 1 + 1/3*x + 7/15*x^2 + 9/35*x^3 + ...

MAPLE

# double factorial of odd "l" df := proc(l) local n; n := iquo(l, 2); RETURN( factorial(l)/2^n/factorial(n)); end: x := 1; for n from 1 to 15 do if n mod 2 = 0 then x := 2*n*x+df(2*n-1); else x := 2*n*x-df(2*n-1); fi; print(x); od; quit

MATHEMATICA

a[n_] := a[n] = (-1)^n*(2n - 1)!! + 2n*a[n - 1]; a[0] = 1; Table[ a[n], {n, 0, 14}] (* Jean-Fran├žois Alcover, Dec 01 2011, after R. J. Mathar *)

a[ n_] := If[ n < 0, 0, (2 n + 1)!! Hypergeometric2F1[ -n, 1/2, 3/2, 2]]; (* Michael Somos, Apr 20 2018 *)

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / ((1 - 2 x) Sqrt[1 + 2 x]), {x, 0, n}]]; (* Michael Somos, Apr 20 2018 *)

RecurrenceTable[{a[0]==a[1]==1, a[n+1]==a[n]+2n(2n+1)a[n-1]}, a, {n, 20}] (* Harvey P. Dale, Jul 27 2019 *)

PROG

(Haskell)

a003148 n = a003148_list !! n

a003148_list = 1 : 1 : zipWith (+) (tail a003148_list)

                          (zipWith (*) (tail a002943_list) a003148_list)

-- Reinhard Zumkeller, Nov 22 2011

(PARI) Vec(serlaplace(1/(sqrt(1+2*x + O(x^20))*(1-2*x)))) \\ Andrew Howroyd, Feb 05 2018

CROSSREFS

Contribution from Johannes W. Meijer, Nov 23 2009: (Start)

Appears in A167552, A167565, A167580 and A167591.

Equals A049606*A123746.

(End)

Cf. A002943.

Sequence in context: A193257 A173193 A196323 * A033910 A196647 A274579

Adjacent sequences:  A003145 A003146 A003147 * A003149 A003150 A003151

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

a(16)-a(20) from Andrew Howroyd, Feb 05 2018

STATUS

approved

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Last modified November 16 22:05 EST 2019. Contains 329208 sequences. (Running on oeis4.)