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A100666
Sum_{k>=1} k^n/Catalan(k) rounded to nearest integer.
0
3, 3, 7, 21, 79, 357, 1879, 11277, 75967, 567381, 4652071, 41534493, 401057935, 4164175845, 46260731384, 547489559470, 6876483788377, 91352567576937, 1279774932585453, 18855298837939164, 291449116254193528
OFFSET
0,1
COMMENTS
The exact values are conjectured to be close to integers, but there is no sound basis for it as yet. The coincidence for the first 10 instances is however intriguing.
FORMULA
Sum[k^n/((2k)!/k!/(k+1)!), {k, \[Infinity]}]==Sum[Hypergeometric2F1[m+1, m+2, m+1/2, 1/4]StirlingS2[n, m]/(2m-1)!!/2^m(m+1)!m!, {m, 1, n}]; see Formula. Hypergeometric2F1[m, m+1, m-1/2, 1/4] equal to h[a_]:=h[a]=Apart[(4*(-3+2*a)*((-5+2*a)*h[ -2+a]-(-4+a)*h[ -1+a]))/(3*(-1+a)*a)]; h[1]:=2+(4*Pi)/(9*Sqrt[3]); h[0]:=1;
EXAMPLE
n float(n) Exact(n)
0 2.806133 2 + (4*Pi)/(9*Sqrt[3])
1 3.074844 2 + (16*Pi)/(27*Sqrt[3])
2 6.995495 (2*(567 + 52*Sqrt[3]*Pi))/243
3 20.986486 14 + (104*Pi)/(27*Sqrt[3])
4 79.000346 158/3 + (392*Pi)/(27*Sqrt[3])
MATHEMATICA
Hypergeometric2F1[a, a+1, a-1/2, 1/4] equals h[a_]:=h[a]=Apart[(4*(-3+2*a)*((-5+2*a)*h[ -2+a]-(-4+a)*h[ -1+a]))/(3*(-1+a)*a)]; h[1]:=2+(4*Pi)/(9*Sqrt[3]); h[0]:=1; Table[Round[Sum[h[m+1] StirlingS2[n, m]/(2m-1)!!/2^m (m+1)!m!, {m, 0, n}]], {n, 14}]
CROSSREFS
Sequence in context: A097334 A214496 A046631 * A262375 A232368 A333924
KEYWORD
easy,nonn
AUTHOR
Wouter Meeussen, Dec 05 2004
STATUS
approved