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A006134
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a(n) = Sum_{ k=0..n } binomial(2*k,k).
(Formerly M2811)
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39
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1, 3, 9, 29, 99, 351, 1275, 4707, 17577, 66197, 250953, 956385, 3660541, 14061141, 54177741, 209295261, 810375651, 3143981871, 12219117171, 47564380971, 185410909791, 723668784231, 2827767747951, 11061198475551, 43308802158651
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Simpler definition from Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 05 2006
Comment from N. J. A. Sloane (njas(AT)research.att.com), Jan 21 2009: The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)/B^k gives A006134 for B=1, A082590 (B=2), A132310 (B=3), A002457 (B=4), A144635 (B=5).
T(n+1,1) from table A045912 of characteristic polynomial of negative Pascal matrix.
p divides a((p-3)/2) for p=11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167..=A097933. Also primes congruent to {1, 2, 3, 11} mod 12 or primes p such that 3 is a square mod p (excluding 2 and 3) A038874. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 05 2006
Partial sums of the even central binomial coefficients. For p prime >=5, a(p-1) = 1 or -1 (mod p) according as p = 1 or -1 (mod 3) (see Pan and Sun link). - David Callan (callan(AT)stat.wisc.edu), Nov 29 2007
a(n)=Sum_{ k=0..n } b(k)*binomial(n+k,k), where b(k)=0 for n-k == 2 (mod 3), b(k)=1 for n-k == 0 or 1 (mod 6), and b(k)=-1 for n-k== 3 or 4 (mod 6). - Alzhekeyev Ascar M. - Jan 19 2012
a(n)=Sum_{ k=0..n-1 } c(k)*binomial(2n,k) + binomial(2n,n), where c(k)=0 for n-k == 0 (mod 3), c(k)=1 for n-k== 1 (mod 3), and c(k)=-1 for n-k==2 (mod 3). - Alzhekeyev Ascar M. - Jan 19 2012
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REFERENCES
| W. F. Lunnon, "The Pascal matrix", Fib. Quart. vol. 15 (1977) pp. 201-204.
M. Petkovsek et al., A=B, Peters, 1996, p. 22.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Hao Pan and Zhi-Wei Sun, A combinatorial identity with application to Catalan numbers .
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FORMULA
| a(n) = Sum[((2k)!/(k!)^2),{k,0,n}]. a(n) = A066796(n) + 1, n>0. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 05 2006
G.f.: 1/((1-x)*sqrt(1-4*x)).
Recurrence: (n+2)*a(n+2) - (5*n+8)*a(n+1) + 2*(2*n+3)*a(n) = 0 [Emanuele Munarini, Mar 15 2011]
a(n) = C(2n,n) * Sum_{k=0..2n} (-1)^k*trinomial(n,k)/C(2n,k) where trinomial(n,k) = [x^k] (1 + x + x^2)^n. E.g. a(2) = C(4,2)*(1/1 - 2/4 + 3/6 - 2/4 + 1/1) = 6*(3/2) = 9 ; a(3) = C(6,3)*(1/1 - 3/6 + 6/15 - 7/20 + 6/15 - 3/6 + 1/1) = 20*(29/20) = 29. - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 21 2007
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EXAMPLE
| Cf. A006135, A006136, A045912. Differences give A000984.
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MAPLE
| A006134 := proc(n) sum(binomial(2*k, k), k=0..n); end;
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MATHEMATICA
| Table[Sum[((2k)!/(k!)^2), {k, 0, n}], {n, 0, 50}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 05 2006
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PROG
| (MATLAB) n=10; x=pascal(n); trace(x)
(PARI) a(n)=if(n<0, 0, polcoeff(charpoly(matrix(n+1, n+1, i, j, -binomial(i+j-2, i-1))), 1))
(PARI) {a(n)=binomial(2*n, n)*sum(k=0, 2*n, (-1)^k*polcoeff((1+x+x^2)^n, k)/binomial(2*n, k))} - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 21 2007
(Maxima) makelist(sum(binomial(2*k, k), k, 0, n), n, 0, 12); [Emanuele Munarini, Mar 15 2011]
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CROSSREFS
| Cf. A000984, A066796, A097933, A038874.
Cf. A132310.
Sequence in context: A151030 A066331 A099780 * A074526 A148940 A169781
Adjacent sequences: A006131 A006132 A006133 * A006135 A006136 A006137
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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