A292499 a(n) equals the sum of the Gaussian binomial coefficients [n,k] at q=n for n>=0.

1, 2, 5, 28, 529, 42176, 16649389, 33736398032, 409516908802369, 28081351726592246272, 13485189809930561625032701, 39060769254069395008311334483648, 909068834019126312744002601418684280593, 133895756625158337189123339374443720098444148736, 179003735935372017689180463552762995401182868750095031853

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..50

Formula

a(n) = 2 + Sum_{m=1..n-1} Product_{k=1..m} (n^(n-k+1) - 1)/(n^k - 1) for n>0, with a(0) = 1.

Example

Terms equal the row sums of the triangle:
1;
1, 1;
1, 3, 1;
1, 13, 13, 1;
1, 85, 357, 85, 1;
1, 781, 20306, 20306, 781, 1;
1, 9331, 2072815, 12485095, 2072815, 9331, 1;
1, 137257, 336416907, 16531644851, 16531644851, 336416907, 137257, 1;
1, 2396745, 79783113865, 40928737412745, 327499862955657, 40928737412745, 79783113865, 2396745, 1; ...
in which row n lists the Gaussian binomial coefficients [n,k] at q=n.

Maple

b:= proc(n, h, m) option remember; `if`(n=0, 1,
      h^m*b(n-1, h, m)+b(n-1, h, m+1))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 08 2021

Mathematica

a[n_] := Sum[QBinomial[n, k, n], {k, 0, n}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Sep 14 2022 *)

Prog

(PARI) {a(n) = if(n==0, 1, 2 + sum(m=1, n-1, prod(k=1, m, (n^(n-k+1)-1)/(n^k-1) ) ))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Sequence in context: A138293 A316972 A068069 * A306893 A105787 A110497

Adjacent sequences: A292496 A292497 A292498 * A292500 A292501 A292502

Keyword

nonn

Author

Paul D. Hanna, Sep 19 2017

Status

approved