A382434 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^4.

1, 1, 3, 33, 195, 1763, 15623, 156257, 1630947, 17911299, 203739015, 2389928995, 28749060871, 353362388551, 4424242664975, 56290517376737, 726355164976547, 9490129871680355, 125375330053632455, 1672895457018337859, 22522481793315373319, 305695116823973096519

Offset: 0

Author

Seiichi Manyama, Mar 25 2025

Keywords

• nonn

Crossrefs

Cf. A131428, A382435.

Cf. A080233, A129123, A156644.

Programs

Maple

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> 2*add(b(n, n-2*j)^4, j=0..n/2)-1:
seq(a(n), n=0..21);  # Alois P. Heinz, Mar 25 2025

PARI

a(n) = sum(k=0, n, (binomial(n, k)-binomial(n, k-1))^4);

Python

from math import comb
def A382434(n): return (sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**4 for j in range((n>>1)+1))<<1)-1 # Chai Wah Wu, Mar 25 2025

Formula

a(n) = Sum_{k=0..n} A080233(n,k)^4 = Sum_{k=0..n} A156644(n,k)^4.

a(n) = 2 * A129123(n) - 1.

D-finite with recurrence n*(n+1)^3*a(n) -2*n*(11*n^3-17*n^2+5*n+5)*a(n-1) -4*(n-1)*(70*n^3-365*n^2+527*n-162)*a(n-2) +8*(n-2)*(584*n^3-5020*n^2+14111*n-13059)*a(n-3) +1344*(4*n-11)*(4*n-13)*(-3+n)^2*a(n-4) +9*(2875*n^4-33975*n^3+149945*n^2-293541*n+215336)=0. - R. J. Mathar, Mar 31 2025