|
Search: id:A006134
|
|
|
| A006134 |
|
T(n+1,1) from table A045912 of characteristic polynomial of negative Pascal matrix.
(Formerly M2811)
|
|
+0
27
|
|
| 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; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
From characteristic polynomial of 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 Primes p > 3 such that p divides 3^(p-1)/2 - 1. 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
|
|
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.
|
|
LINKS
|
Hao Pan and Zhi-Wei Sun,A combinatorial identity with application to Catalan numbers .
|
|
FORMULA
|
C(2n,n) + C(2n-2,n-1) + ... + C(2k,k) + ... + C(2,1) + C(0,0), where C(2k,k)=(2k)!/(k!)^2 - central binomial coefficients A000984[k]. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 05 2006
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
|
|
EXAMPLE
|
Cf. A006135, A006136, A045912. Differences give A000984.
|
|
MAPLE
|
A006134 := proc(n) sum(binomial(2*k, k), k=0..n); end;
|
|
MATHEMATICA
|
Table[Sum[((2k)!/(k!)^2), {k, 0, n}], {n, 0, 50}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 05 2006
|
|
PROGRAM
|
(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
|
|
CROSSREFS
|
Cf. A000984, A066796, A097933, A038874.
Cf. A132310.
Sequence in context: A001893 A066331 A099780 this_sequence A074526 A079319 A112532
Adjacent sequences: A006131 A006132 A006133 this_sequence A006135 A006136 A006137
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
njas
|
|
|
Search completed in 0.003 seconds
|
