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A241755
A finite sum of products of binomial coefficients: Sum_(m=0..n) binomial(-1/4, m)^2*binomial(-1/4, n-m)^2 (C. C. Grosjean's problem, numerators).
1
1, 1, 27, 125, 42875, 250047, 12326391, 78953589, 266468362875, 1795828623875, 98540708249269, 685638992559339, 308969245276647319, 2197380271937921875, 126096314555551359375, 911218671317138401125, 27146115437208870107914875
OFFSET
0,3
COMMENTS
Quoted from SIAM: This sum arises from the calculation of the shift of the frequency of an electromagnetic transverse magnetic wave-mode caused by a small metallic cylinder in a resonant cavity.
REFERENCES
E. S. Andersen and M. E. Larsen. A finite sum of products of binomial coefficients, Problem 92-18, by C. C. Grosjean, Solution. SIAM Rev. 35 (1993), 645-646.
LINKS
P. Flajolet, B. Salvy, and Helmut Prodinger, A Finite Sum of Products of Binomial Coefficients, Problem 92-18 by C. C. Grosjean, Solution. SIAM Rev. 35 (1993), 645-646.
C. C. Grosjean, Problem no. 92-18, SIAM Rev. 34 (1992), p. 649.
M. E. Larsen, Summa Summarum, page 114.
FORMULA
GAMMA(3/4)^2 * 4F3(1/4, 1/4, -n, -n; 1, 3/4-n, 3/4-n; 1)/(GAMMA(3/4-n)^2*GAMMA(n+1)^2).
binomial(2n, n)^2*binomial(n-1/2, 2n)*(-1/4)^n.
EXAMPLE
1, 1/8, 27/512, 125/4096, 42875/2097152, 250047/16777216, ...
MATHEMATICA
a[n_] := Binomial[2*n, n]^2*Binomial[n-1/2, 2*n]*(-1/4)^n; Table[a[n]//Numerator, {n, 0, 20}]
CROSSREFS
Cf. A241756.
Sequence in context: A092770 A253103 A321486 * A044359 A044740 A026917
KEYWORD
nonn,frac
AUTHOR
STATUS
approved