0,2

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. F. Lunnon "The Pascal matrix", Fib. Quart. vol. 15 (1977) pp. 201-204.

M. Petkovsek et al., A=B, Peters, 1996, p. 22.

Hao Pan and Zhi-Wei Sun, A combinatorial identity with application to Catalan numbers .

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)).

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

Cf. A006135, A006136, A045912. Differences give A000984.

A006134 := proc(n) sum(binomial(2*k, k), k=0..n); end;

restart:a:= proc(n) option remember; if n=0 then 1 else add((binomial (2*j, j)), j=0..n-1) fi end: seq (a(n), n=1..25); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 29 2009]

Table[Sum[((2k)!/(k!)^2), {k, 0, n}], {n, 0, 50}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 05 2006

(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

Cf. A000984, A066796, A097933, A038874.

Cf. A132310.

Sequence in context: A151030 A066331 A099780 this_sequence A074526 A148940 A162998

Adjacent sequences: A006131 A006132 A006133 this_sequence A006135 A006136 A006137

nonn

N. J. A. Sloane (njas(AT)research.att.com).