A345458 a(n) = Sum_{k=0..n} binomial(5*n+4,5*k).

1, 127, 3004, 107883, 3321891, 107746282, 3431847189, 109996928003, 3517929664756, 112595619434887, 3602817278095399, 115292842751246298, 3689341137121931721, 118059247217851456567, 3777892242010882603884, 120892592433742197034643

Offset: 0

Author

Seiichi Manyama, Jun 20 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (21,353,-32).

Crossrefs

Sum_{k=0..n} binomial(b*n+c,b*k): A070782 (b=5,c=0), A345455 (b=5,c=1), A345456 (b=5,c=2), A345457 (b=5,c=3), this sequence (b=5,c=4).

Cf. A139398.

Programs

Mathematica

a[n_] := Sum[Binomial[5*n + 4, 5*k], {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Jun 20 2021 *)
Total/@Table[Binomial[5n+4,5k],{n,0,20},{k,0,n}] (* or *) LinearRecurrence[{21,353,-32},{1,127,3004},30] (* Harvey P. Dale, Oct 29 2023 *)

PARI

a(n) = sum(k=0, n, binomial(5*n+4, 5*k));

PARI

my(N=20, x='x+O('x^N)); Vec((1+106*x-16*x^2)/((1-32*x)*(1+11*x-x^2)))

Formula

G.f.: (1 + 106*x - 16*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)).

a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>2.

a(n) = A139398(5*n+4).

a(n) = 2^(5*n + 5)/10 + ((2015 - 901*sqrt(5))/phi^(5*n) - (35 + sqrt(5))*(-1)^n*phi^(5*n)) / (10*(41*sqrt(5)-90)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 20 2021