A064095 Row sums of triangle A064094.

1, 2, 3, 5, 11, 34, 142, 753, 4826, 36028, 305133, 2879841, 29909422, 338479430, 4139716659, 54339861531, 761150445735, 11322139144240, 178116143657890, 2952831190016239, 51423702126549167, 938126972940647198, 17883424301972473340, 355435808475002747565, 7350551776003412371185

Offset: 0

Author

Wolfdieter Lang, Sep 13 2001

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..475

Crossrefs

Cf. A064094.

Programs

Magma

function A064094(n,k)
  if k eq 0 or k eq n then return 1;
  else return (&+[(n-k-j)*Binomial(n-k-1+j, j)*k^j: j in [0..n-k-1]])/(n-k);
  end if;
end function;
A064095:= func< n | (&+[A064094(n,k): k in [0..n]]) >;
[A064095(n): n in [0..30]]; // G. C. Greubel, Sep 27 2024

Mathematica

A064094[n_, k_]:= If[k==0 || k==n, 1, Sum[(n-k-j)*Binomial[n-k-1+j, j]*k^j, {j,0,n-k-1}]/(n-k) ];
A064095[n_]:= Sum[A064094[n,k], {k,0,n}];
Table[A064095[n], {n,0,30}] (* G. C. Greubel, Sep 27 2024 *)

PARI

T(n, k)= if (n==k, 1, sum(i=0, n-k-1, (n-k-i)*binomial(n-k-1+i, i)*(k^i)/(n-k))); \\ A064094
a(n) = sum(k=0, n, T(n,k));

SageMath

def A064094(n,k):
    if (k==0 or k==n): return 1
    else: return sum((n-k-j)*binomial(n-k-1+j, j)*k^j for j in range(n-k))//(n-k)
def A064095(n): return sum(A064094(n,k) for k in range(n+1))
[A064095(n) for n in range(31)] # G. C. Greubel, Sep 27 2024

Extensions

More terms from Michel Marcus, Oct 28 2022