A198059 a(n) = Sum_{k=1..n} binomial(2*k, n-k)^2 * n/k.

1, 9, 28, 121, 496, 2100, 9017, 38969, 169975, 744984, 3282005, 14513236, 64394500, 286519305, 1277975053, 5712392313, 25581765122, 114754116351, 515530099946, 2319115721576, 10445215621547, 47096725844837, 212569226371737, 960306310551860, 4341968468524371

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

Logarithmic derivative of A197601.

L.g.f.: Sum_{n>=1} [Sum_{k=0..2*n} C(2*n,k)^2 *x^k] *x^n/n.

L.g.f.: Sum_{n>=1} (1-x)^(4*n+1) *[Sum_{k>=0} C(2*n+k,k)^2 *x^k] *x^n/n.

G.f.: sqrt((1 + (15*x^3+2*x^2-x+3)/w - (x^3-2*x^2+x-4)/sqrt(w))/2) - 2 where w = (x^3-2*x^2-3*x-1)*(x^3-2*x^2+5*x-1). - Mark van Hoeij, May 06 2013

Example

L.g.f.: L(x) = x + 9*x^2/2 + 28*x^3/3 + 121*x^4/4 + 496*x^5/5 + 2100*x^6/6 + ...
where
exp(L(x)) = 1 + x + 5*x^2 + 14*x^3 + 52*x^4 + 187*x^5 + 708*x^6 + ... + A197601(n)*x^n + ...
The l.g.f. equals the series:
L(x) = (1 + 2^2*x + x^2)*x
+ (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)*x^2/2
+ (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)*x^3/3
+ (1 + 8^2*x + 28^2*x^2 + 56^2*x^3 + 70^2*x^4 + 56^2*x^5 + 28^2*x^6 + 8^2*x^7 + x^8)*x^4/4
+ (1 + 10^2*x + 45^2*x^2 + 120^2*x^3 + 210^2*x^4 + 252^2*x^5 + 210^2*x^6 + 120^2*x^7 + 45^2*x^8 + 10^2*x^9 + x^10)*x^5/5 + ...
which involves the squares of the coefficients in even powers of (1+x).
Also,
L(x) = (1-x)^5*(1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 15^2*x^4 + ...)*x
+ (1-x)^9*(1 + 5^2*x + 15^2*x^2 + 35^2*x^3 + 70^2*x^4 + ...)*x^2/2
+ (1-x)^13*(1 + 7^2*x + 28^2*x^2 + 84^2*x^3 + 210^2*x^4 + ...)*x^3/3
+ (1-x)^17*(1 + 9^2*x + 45^2*x^2 + 165^2*x^3 + 495^2*x^4 + ...)*x^4/4
+ (1-x)^21*(1 + 11^2*x + 66^2*x^2 + 286^2*x^3 + 1001^2*x^4 + ...)*x^5/5 + ...
which involves the squares of the coefficients in odd powers of 1/(1-x).

Maple

w := (x^3-2*x^2-3*x-1)*(x^3-2*x^2+5*x-1);
sqrt((1 + (15*x^3+2*x^2-x+3)/w - (x^3-2*x^2+x-4)/sqrt(w))/2) - 2;
series(%, x=0, 30); # Mark van Hoeij, May 06 2013

Mathematica

Table[Sum[Binomial[2k, n-k]^2 n/k, {k, n}], {n, 30}] (* Harvey P. Dale, Oct 25 2011 *)

Prog

(PARI) {a(n)=n*sum(k=1, n, binomial(2*k, n-k)^2/k)}
(PARI) {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(2*m, k)^2 *x^k)*x^m/m)+x*O(x^n), n)}
(PARI) {a(n)=n*polcoeff(sum(m=1, n, (1-x+x*O(x^n))^(4*m+1) *sum(k=0, n-m+1, binomial(2*m+k, k)^2 *x^k)*x^m/m+x*O(x^n)), n)}

Crossrefs

Cf. A197601 (exp).

Sequence in context: A294287 A349547 A085292 * A181090 A073706 A226976

Adjacent sequences: A198056 A198057 A198058 * A198060 A198061 A198062

Keyword

nonn

Author

Paul D. Hanna, Oct 20 2011

Status

approved