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A206851 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2)*x^k = Sum_{n>=1} a(n)*x^n/n. 7
1, 3, 7, 15, 231, 2763, 37773, 3347359, 145164760, 15115517783, 5300285945494, 841490209145991, 700215432847179640, 821522962294608211319, 580955012898669141073842, 3240132942509582109732641935, 12114306457535986210506222037102 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Equals the logarithmic derivative of A206850.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..94

FORMULA

a(n) = n * Sum_{k=0..[n/2]} binomial((n-k)^2, k^2)/(n-k).

Limit n->infinity a(n)^(1/n^2) = (1-2*r)^r / r^(2*r) = 1.2915356633069917227119166349..., where r = A323778 = 0.365498498219858044579... is the root of the equation (1-r)^(2-2*r) * r^(2*r) = 1-2*r. - Vaclav Kotesovec, Mar 03 2014

EXAMPLE

L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 231*x^5/5 + 2763*x^6/6 +...

where exponentiation yields the g.f. of A206850:

exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 56*x^5 + 522*x^6 + 5972*x^7 +...

By definition, the l.g.f. equals the series:

L(x) = (C(1,0) + C(1,1)*x)*x

+ (C(4,0) + C(4,1)*x + C(4,4)*x^2)*x^2/2

+ (C(9,0) + C(9,1)*x + C(9,4)*x^2 + C(9,9)*x^3)*x^3/3

+ (C(16,0) + C(16,1)*x + C(16,4)*x^2 + C(16,9)*x^3 + C(16,16)*x^4)*x^4/4

+ (C(25,0) + C(25,1)*x + C(25,4)*x^2 + C(25,9)*x^3 + C(25,16)*x^4 + C(25,25)*x^5)*x^5/5 +...

More explicitly,

L(x) = (1 + 1*x)*x + (1 + 4*x + 1*x^2)*x^2/2

+ (1 + 9*x + 126*x^2 + 1*x^3)*x^3/3

+ (1 + 16*x + 1820*x^2 + 11440*x^3 + 1*x^4)*x^4/4

+ (1 + 25*x + 12650*x^2 + 2042975*x^3 + 2042975*x^4 + 1*x^5)*x^5/5

+ (1 + 36*x + 58905*x^2 + 94143280*x^3 + 7307872110*x^4 + 600805296*x^5 + 1*x^6)*x^6/6 +...

MATHEMATICA

Table[n*Sum[Binomial[(n-k)^2, k^2]/(n-k), {k, 0, Floor[n/2]}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)

PROG

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

(PARI) {a(n)=n*sum(k=0, n\2, binomial((n-k)^2, k^2)/(n-k))}

for(n=1, 20, print1(a(n), ", "))

CROSSREFS

Cf. A206850 (exp), A206847, A206849, A123165.

Sequence in context: A246719 A077775 A197594 * A033089 A175878 A180503

Adjacent sequences:  A206848 A206849 A206850 * A206852 A206853 A206854

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 15 2012

STATUS

approved

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Last modified July 8 19:20 EDT 2020. Contains 335524 sequences. (Running on oeis4.)