A301308 G.f.: Sum_{n>=0} x^n * ( (1+x)^n + (1-x)^n )^n / 2^n.

1, 1, 1, 1, 3, 10, 26, 78, 311, 1200, 4626, 20831, 98953, 468700, 2399950, 13084381, 72319603, 418017254, 2545312690, 15862538760, 102062357538, 684056780835, 4705498914532, 33243430090585, 242674907474251, 1817152315653253, 13936783521933730, 109852260904263059, 886654177280345126, 7315920883507888061, 61812165690051526144

Offset

0,5

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

G.f.: Sum_{n>=0} x^n * [ Sum_{k=0..[n/2]} binomial(n,2*k) * x^(2*k) ]^n.

G.f.: Sum_{n>=0} x^n * (1+x)^(n^2) * Sum_{k=0..n} C(n,k) * ((1-x)/(1+x))^(n*k) / 2^n.

Example

G.f.: A(x) = 1 + x + x^2 + x^3 + 3*x^4 + 10*x^5 + 26*x^6 + 78*x^7 + 311*x^8 + 1200*x^9 + 4626*x^10 + 20831*x^11 + 98953*x^12 + ...
such that
A(x) = 1  +  x*((1+x) + (1-x))/2  +  x^2*((1+x)^2 + (1-x)^2)^2/2^2  +  x^3*((1+x)^3 + (1-x)^3)^3/2^3  +  x^4*((1+x)^4 + (1-x)^4)^4/2^4  +  x^5*((1+x)^5 + (1-x)^5)^5/2^5  +  x^6*((1+x)^6 + (1-x)^6)^6/2^6  + ...
Equivalently,
A(x) = 1 + x + x^2*(1 + x^2)^2 + x^3*(1 + 3*x^2)^3 + x^4*(1 + 6*x^2 + x^4)^4 + x^5*(1 + 10*x^2 + 5*x^4)^5 + x^6*(1 + 15*x^2 + 15*x^4 + x^6)^6 + x^7*(1 + 21*x^2 + 35*x^4 + 7*x^6)^7 + x^8*(1 + 28*x^2 + 70*x^4 + 28*x^6 + x^8)^8 + ...

Prog

(PARI) {a(n) = my(A=1); A = sum(m=0, n, x^m*((1+x)^m + (1-x)^m +x*O(x^n))^m/2^m ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A269965 A375140 A110158 * A196232 A282282 A105660

Adjacent sequences: A301305 A301306 A301307 * A301309 A301310 A301311

Keyword

nonn

Author

Paul D. Hanna, Mar 18 2018

Status

approved