login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A207140 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2,k^2). 1
1, 2, 10, 407, 56746, 30771252, 115106662819, 1446405270234360, 53819202633553797290, 12313337704248075967333334, 12373818231445938048765251252260, 33156027144321617106970597265032233270, 409476940913917468665022448013012674533441891 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Ignoring initial term a(0), equals the logarithmic derivative of A207139.

LINKS

Table of n, a(n) for n=0..12.

FORMULA

Limit n->infinity a(n)^(1/n^2) = 2. - Vaclav Kotesovec, Mar 03 2014

EXAMPLE

L.g.f.: L(x) = 2*x + 10*x^2/2 + 407*x^3/3 + 56746*x^4/4 + 30771252*x^5/5 +...

where exponentiation equals the g.f. of A207139:

exp(L(x)) = 1 + 2*x + 7*x^2 + 147*x^3 + 14481*x^4 + 6183605*x^5 +...

By definition, the initial terms begin: a(0) = 1;

a(1) = C(1,0)*C(1,0), + C(1,1)*C(1,1);

a(2) = C(2,0)*C(4,0), + C(2,1)*C(4,1), + C(2,2)*C(4,4);

a(3) = C(3,0)*C(9,0), + C(3,1)*C(9,1), + C(3,2)*C(9,4), + C(3,3)*C(9,9);

a(4) = C(4,0)*C(16,0), + C(4,1)*C(16,1), + C(4,2)*C(16,4), + C(4,3)*C(16,9), + C(4,4)*C(16,16); ...

which is evaluated as:

a(1) = 1*1 + 1*1 = 2;

a(2) = 1*1 + 2*4 + 1*1 = 10;

a(3) = 1*1 + 3*9 + 3*126 + 1*1 = 407;

a(4) = 1*1 + 4*16 + 6*1820 + 4*11440 + 1*1 = 56746;

a(5) = 1*1 + 5*25 + 10*12650 + 10*2042975 + 5*2042975 + 1*1 = 30771252;

a(6) = 1*1 + 6*36 + 15*58905 + 20*94143280 + 15*7307872110 + 6*600805296 + 1*1 = 115106662819; ...

MATHEMATICA

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

PROG

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

for(n=0, 16, print1(a(n), ", "))

CROSSREFS

Cf. A207139 (exp), A206851, A207136, A207138, A167009.

Sequence in context: A206152 A261007 A013034 * A059723 A265627 A112449

Adjacent sequences:  A207137 A207138 A207139 * A207141 A207142 A207143

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 15 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 22 18:24 EST 2020. Contains 332148 sequences. (Running on oeis4.)