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

%I A382434 #19 Mar 31 2025 06:30:46
%S A382434 1,1,3,33,195,1763,15623,156257,1630947,17911299,203739015,2389928995,
%T A382434 28749060871,353362388551,4424242664975,56290517376737,
%U A382434 726355164976547,9490129871680355,125375330053632455,1672895457018337859,22522481793315373319,305695116823973096519
%N A382434 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^4.
%F A382434 a(n) = Sum_{k=0..n} A080233(n,k)^4 = Sum_{k=0..n} A156644(n,k)^4.
%F A382434 a(n) = 2 * A129123(n) - 1.
%F A382434 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
%p A382434 b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
%p A382434      `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
%p A382434     end:
%p A382434 a:= n-> 2*add(b(n, n-2*j)^4, j=0..n/2)-1:
%p A382434 seq(a(n), n=0..21);  # _Alois P. Heinz_, Mar 25 2025
%o A382434 (PARI) a(n) = sum(k=0, n, (binomial(n, k)-binomial(n, k-1))^4);
%o A382434 (Python)
%o A382434 from math import comb
%o A382434 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
%Y A382434 Cf. A131428, A382435.
%Y A382434 Cf. A080233, A129123, A156644.
%K A382434 nonn
%O A382434 0,3
%A A382434 _Seiichi Manyama_, Mar 25 2025