A084625 Binomial transform of A084624.

1, 3, 8, 21, 55, 143, 366, 919, 2265, 5491, 13125, 31000, 72485, 168042, 386709, 884161, 2009742, 4543830, 10222264, 22891099, 51041560, 113359224, 250839510, 553173006, 1216070081, 2665518207, 5826533103, 12703217438, 27628250142

Offset: 0

Author

Paul Barry, Jun 01 2003

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A084624, A084626, A084627, A084628, A084630, A084631.

Programs

Magma

A084625:= func< n | (&+[Binomial(n,j)*Floor(Binomial(j+5,3)/10): j in [0..n]]) >;
[A084625(n): n in [0..50]]; // G. C. Greubel, Mar 24 2023

Mathematica

a[n_]:= a[n]= 2^n +Sum[Binomial[n,j]*Floor[j*(j^2+12*j+47)/60], {j,0, n}];
Table[a[n], {n,0,50}] (* G. C. Greubel, Mar 24 2023 *)

SageMath

def A084625(n): return sum(binomial(n,j)*(binomial(j+5,3)//10) for j in range(n+1))
[A084625(n) for n in range(51)] # G. C. Greubel, Mar 24 2023

Formula

a(n) = Sum_{k=0..n} C(n, k)*floor(C(k+5, 5)/C(k+2, 2)).

a(n) = 2^n + Sum_{k=0..n} binomial(n,k)*floor(k*(k^2 +12*k +47)/60). - G. C. Greubel, Mar 24 2023