%I #33 Oct 25 2024 09:54:29

%S 1,4,28,364,9100,445900,43252300,8347693900,3213862151500,

%T 2471459994503500,3798634011551879500,11673202317498925703500

%N Positive numbers appearing in the third column of A155103.

%F From _Tristan Cam_, Oct 02 2024: (Start)

%F a(1) = 1, a(n) = a(n-1)*(1+3*2^(n-2)) (conjectured).

%F a(n) = Product_{k=1..n-1} 1+3*2^(k-1) = QPochhammer[-3, 2, n-1]. (conjectured). (End)

%p A155102 := proc(n,k) if n = k then 1 ; elif n =2*k then -k-1 ; else 0; end if; end proc:

%p A155103 := proc(amx) a := array(1..amx,1..amx) ; a[1,1] := 1/A155102(1,1) ;

%p for r from 1 to amx do

%p for c from 1 to r-1 do a[c,r] := 0 ; end do:

%p a[r,r] := 1/A155102(r,r) ;

%p for c from r-1 to 1 by -1 do a[r,c] := -add(a[cp,c]*A155102(r,cp),cp=c..r-1)/A155102(r,r) ;

%p if c = 3 and a[r,c] <> 0 then print( a[r,c]) ; end if;

%p end do:

%p end do:

%p return ;

%p end proc:

%p A155103(290) ; # _R. J. Mathar_, Dec 07 2010

%o (PARI) \\ after _R. J. Mathar_

%o T(n,k)=if(n==k,1,if(n==2*k,-(k+1))); \\ from A155102

%o \\ First term = 1 omitted

%o a155103(upto) = my(m=3*2^upto, a=matid(m)); for(r=1, m, forstep(c=r-1, 1, -1, a[r,c]=-sum(cp=c, r-1, a[cp,c]*T(r,cp)); if(c==3 && a[r,c]!=0, print1(a[r,c],", "))));

%o a155103(8) \\ _Hugo Pfoertner_, Oct 03 2024

%Y Cf. A155103.

%K nonn,more,changed

%O 1,2

%A _Mats Granvik_, Jan 20 2009

%E Two more terms from _R. J. Mathar_, Dec 07 2010

%E a(8)-a(12) from _Hugo Pfoertner_, Oct 02 2024