login
a(n) equals the sum of the Gaussian binomial coefficients [n,k] at q=n for n>=0.
1

%I #11 Sep 14 2022 08:25:27

%S 1,2,5,28,529,42176,16649389,33736398032,409516908802369,

%T 28081351726592246272,13485189809930561625032701,

%U 39060769254069395008311334483648,909068834019126312744002601418684280593,133895756625158337189123339374443720098444148736,179003735935372017689180463552762995401182868750095031853

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

%H Paul D. Hanna, <a href="/A292499/b292499.txt">Table of n, a(n) for n = 0..50</a>

%F 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.

%e Terms equal the row sums of the triangle:

%e 1;

%e 1, 1;

%e 1, 3, 1;

%e 1, 13, 13, 1;

%e 1, 85, 357, 85, 1;

%e 1, 781, 20306, 20306, 781, 1;

%e 1, 9331, 2072815, 12485095, 2072815, 9331, 1;

%e 1, 137257, 336416907, 16531644851, 16531644851, 336416907, 137257, 1;

%e 1, 2396745, 79783113865, 40928737412745, 327499862955657, 40928737412745, 79783113865, 2396745, 1; ...

%e in which row n lists the Gaussian binomial coefficients [n,k] at q=n.

%p b:= proc(n, h, m) option remember; `if`(n=0, 1,

%p h^m*b(n-1, h, m)+b(n-1, h, m+1))

%p end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..15); # _Alois P. Heinz_, Aug 08 2021

%t a[n_] := Sum[QBinomial[n, k, n], {k, 0, n}];

%t Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Sep 14 2022 *)

%o (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) ) ))}

%o for(n=0,20,print1(a(n),", "))

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 19 2017