A160565 -id:A160565 - cp's OEIS frontend

A025252 a(n) = (1/2)*s(n+3), where s = A025251.

0, 1, 2, 1, 6, 9, 12, 41, 60, 121, 310, 505, 1162, 2577, 4760, 11089, 23256, 47089, 107274, 223345, 476366, 1061017, 2237796, 4888313, 10745748, 23048169, 50792638, 111180265, 241786898, 534219297, 1170798128, 2570337441, 5684509232, 12503504353, 27613172114

Offset: 1

Clark Kimberling

nonn

			G.f. = x^2 + 2*x^3 + x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 41*x^8 + 60*x^9 + ...

Cf. A025251, A160565.

a[ n_] := SeriesCoefficient[ (1 - x^2 - 4 x^3 - Sqrt[1 - 2 x^2 - 8 x^3 + x^4]) / (4 x^3), {x, 0, n}]; (* Michael Somos, Feb 08 2015 *)

{a(n) = if( n<1, 0, polcoeff( (-sqrt(1 - 2*x^2 - 8*x^3 + x^4 + x^4*O(x^n))) / 4, n+3))};

G.f.: (1 - x^2 - 4*x^3 - sqrt(1 - 2*x^2 - 8*x^3 + x^4)) / (4*x^3). - Michael Somos, Jun 08 2000
(n+3)*a(n) +(n+2)*a(n-1) -2*n*a(n-2) +2*(-5*n+7)*a(n-3) +(-7*n+17)*a(n-4) +(n-4)*a(n-5)=0. - R. J. Mathar, Dec 15 2013
0 = a(n)*(+a(n+1) - 20*a(n+2) - 8*a(n+3) + 7*a(n+5)) +a(n+1)*(+4*a(n+1) + 68*a(n+2) + 40*a(n+3) - 5*a(n+4) - 44*a(n+5)) + a(n+2)*(-8*a(n+2) + 4*a(n+3) + 28*a(n+4) - 8*a(n+5)) + a(n+3)*(+4*a(n+4)) + a(n+4)*(+a(n+5)) for all n>0. - Michael Somos, Feb 08 2015
a(n) = A160565(n) for all n>0. - Michael Somos, Feb 08 2015

A160567 a(n)=4*(a(n-1)a(n-3)-a(n-2)^2)/a(n-4), a(1)=a(2)=a(3)=1, a(4)=-4.

Original entry on oeis.org

1, 1, 1, -4, -20, -144, 704, 34816, 1101824, 12124160, -4499439616, -586464362496, -99309307559936, 33947352788107264, 26466630265219317760, 25787111566764821970944, -7045529183328701486465024

Offset: 0

Paul Barry, May 19 2009

a(n+1) is the Hankel transform of A160565.

Cf. A050512.

A160567 := proc(n)
    option remember;
    if n < 5 then
        op(n,[1,1,1,-4]);
    else
        4*(procname(n-1)*procname(n-3)-procname(n-2)^2)/procname(n-4) ;
    end if;
end proc: # R. J. Mathar, Feb 25 2015

A160568 Diagonal sums of number triangle [k<=n]*C(n,2n-2k)3^(n-k)A000108(n-k).

Original entry on oeis.org

1, 0, 1, 3, 1, 9, 19, 18, 91, 165, 271, 990, 1765, 3843, 11467, 21630, 53299, 140724, 287119, 736101, 1818235, 3982044, 10225117, 24521409, 56584243, 143641017, 341948179, 816095982, 2045559205, 4888806237, 11897144767, 29540684052

Offset: 0

Paul Barry, May 19 2009

Hankel transform is A160569(n+1).

Cf.: A025250, A160565.

G.f.: (1-x^2-sqrt(1-2x^2-12x^3+x^4))/(6*x^3);
G.f.: 1/(1-x^2-3*x^3/(1-x^2-3*x^3/(1-x^2-3*x^3/(1-x^2-3*x^3/(1-... (continued fraction).
a(n)=sum{k=0..floor(n/2), C(n-k,2n-4k)*3^(n-2k)*A000108(n-2k)};
a(n)=sum{k=0..n, C(n-k/2,2(n-k))*3^(n-k)*A000108(n-k)*(1+(-1)^k)/2};
a(n)=sum{k=0..n, C((n+k)/2,2k)*3^k*A000108(k)(1+(-1)^(n-k))/2}.

cp's OEIS Frontend

A025252 a(n) = (1/2)*s(n+3), where s = A025251.

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A160567 a(n)=4*(a(n-1)a(n-3)-a(n-2)^2)/a(n-4), a(1)=a(2)=a(3)=1, a(4)=-4.

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A160568 Diagonal sums of number triangle [k<=n]*C(n,2n-2k)3^(n-k)A000108(n-k).

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