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A065973 Denominators in an asymptotic expansion of Ramanujan. 7
3, 135, 2835, 8505, 12629925, 492567075, 1477701225, 39565450299375, 2255230667064375, 6765692001193125, 7002491221234884375, 21007473663704653125, 441156946937797715625, 56995271759628775870171875 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

REFERENCES

G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge, 1999; Problem 4, p. 616.

B. C. Berndt, Ramanujan's Notebooks II, Springer, 1989; p. 181, Entry 48. See also pp. 184, 193ff.

E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford Univ. Press, 1935; see p. 230, Problem 18.

J. C. W. Marsaglia, The incomplete gamma function and Ramanujan's rational approximation to exp(x), J. Statist. Comput. Simulation, 24 (1986), 163-168.

S. Ramanujan, Collected Papers, edited by G. H. Hardy et al., Cambridge, 1927, pp. 323-324, Question 294.

LINKS

Robert Israel, Table of n, a(n) for n = 0..320 (0 .. 126 from G. C. Greubel and D. Turner)

FORMULA

Define t_n by Sum_{k=0..n-1} n^k/k! + t_n*n^n/n! = exp(n)/2; then t_n ~ 1/3 + 4/(135*n) - 8/(2835*n^2) + ...

Integral_{0..infinity} exp(-x)*(1+x/n)^n dx = exp(n)*Gamma(n+1)/(2*n^n) + 2/3 - 4/(135*n) + 8/(2835*n^2) + 16/(8505*n^3) - 8992/(12629925*n^4) + ...

EXAMPLE

-2/3, 4/135, -8/2835, -16/8505, 8992/12629925, 334144/492567075, -698752/1477701225, ...

MAPLE

# Maple program from N. J. A. Sloane, Jun 23 2011, based on J. Marsaglia's 1986 paper:

a[1]:=1;

M:=20;

for n from 2 to M do

t1:=a[n-1]/(n+1)-add(a[k]*a[n+1-k], k=2..floor(n/2));

if n mod 2 = 1 then t1:=t1-a[(n+1)/2]^2/2; fi;

a[n]:=t1;

od:

s1:=[seq(a[n], n=1..M)]: # This gives A005447/A005446

s2:=[seq(-2^(n+1)*(n+1)!*a[2*n+2], n=0..(M-2)/2)]: # This gives A090804/A065973

map(denom, s2);

MATHEMATICA

Denominator[Table[2^n*(3*n + 2)! * Sum[ Sum[ (-1)^(j + 1)*2^i*StirlingS2[2*n + i + j + 1, j]/((2*n + i + j + 1)!*(2*n - i + 1)!*(i - j)!*(n + i + 1)), {j, 1, i}], {i, 1, 2*n+1}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 20 2015 *)

PROG

(PARI) a(n)=local(A, m); if(n<0, 0, n++; A=vector(m=2*n, k, 1); for(k=2, m, A[k]=(A[k-1]-sum(i=2, k-1, i*A[i]*A[k+1-i]))/(k+1)); denominator(A[m]*2^n*n!)) /* Michael Somos, Jun 09 2004 */

CROSSREFS

Cf. A260306 (numerators), A090804, A005446, A005447.

Sequence in context: A051376 A101721 A173582 * A110973 A136411 A193136

Adjacent sequences:  A065970 A065971 A065972 * A065974 A065975 A065976

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane, Dec 09 2001

EXTENSIONS

Maple program edited by Robert Israel, Dec 15 2015

STATUS

approved

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Last modified February 26 16:54 EST 2021. Contains 341632 sequences. (Running on oeis4.)