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A156058
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a(n) = 5^n * Catalan(n).
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11
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1, 5, 50, 625, 8750, 131250, 2062500, 33515625, 558593750, 9496093750, 164023437500, 2870410156250, 50784179687500, 906860351562500, 16323486328125000, 295863189697265625, 5395152282714843750, 98911125183105468750, 1822047042846679687500
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OFFSET
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0,2
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COMMENTS
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Number of strings of length 2*n of five different types of balanced parentheses.
The number of strings of length 2*n of t different types of balanced parentheses is given by t^n * A000108(n): there are n opening parentheses in the strings, giving t^n choices for the type (the closing parentheses are chosen to match). (End)
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LINKS
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FORMULA
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a(n) is the upper left term in M^n, M = the infinite square production matrix:
5, 5, 0, 0, 0, 0,...
5, 5, 5, 0, 0, 0,...
5, 5, 5, 5, 0, 0,...
5, 5, 5, 5, 5, 0,...
... (End)
D-finite with recurrence (n+1)*a(n) -10*(2*n-1)*a(n-1)=0. - R. J. Mathar, Oct 06 2012
G.f.: 1/(1 - 5*x/(1 - 5*x/(1 - 5*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Apr 19 2017
Sum_{n>=0} 1/a(n) = 410/361 + 600*arctan(1/sqrt(19)) / (361*sqrt(19)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 130/147 - 200*arctanh(1/sqrt(21)) / (147*sqrt(21)). - Amiram Eldar, Jan 25 2022
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MAPLE
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A156058_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 5*(a[w-1]+add(a[j]*a[w-j-1], j=1..w-1)) od; convert(a, list)end: A156058_list(16); # Peter Luschny, May 19 2011
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MATHEMATICA
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Table[5^n CatalanNumber[n], {n, 0, 20}] (* Harvey P. Dale, Mar 13 2011 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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