cp's OEIS Frontend

A132427 Triangle, read by rows of 2n+1 terms, where T(n,k) = T(n,k-1) + T(n-1,k-2) for n>0, 1 with T(n,1)=T(n,0)=T(n-1,2n-2) for n>0 and T(0,0)=1.

%I A132427 #15 Jun 14 2017 00:35:37
%S A132427 1,1,1,2,2,2,3,4,6,6,6,8,10,13,17,23,23,23,29,35,43,53,66,83,106,106,
%T A132427 106,129,152,181,216,259,312,378,461,567,567,567,673,779,908,1060,
%U A132427 1241,1457,1716,2028,2406,2867,3434,3434,3434,4001,4568,5241,6020,6928,7988
%N A132427 Triangle, read by rows of 2n+1 terms, where T(n,k) = T(n,k-1) + T(n-1,k-2) for n>0, 1<k<=2n, with T(n,1)=T(n,0)=T(n-1,2n-2) for n>0 and T(0,0)=1.
%C A132427 Column 0 equals (essentially) column 1 and the rightmost border.
%F A132427 The g.f. of column 0 (A125273) satisfies: G(x) = 1 + x*G( x/(1-x)^2 ) / (1-x).
%F A132427 The central terms (A132428) are the inverse binomial transform of A125273 (offset 1).
%e A132427 Triangle begins:
%e A132427 1;
%e A132427 1, 1, 2;
%e A132427 2, 2, 3, 4, 6;
%e A132427 6, 6, 8, 10, 13, 17, 23;
%e A132427 23, 23, 29, 35, 43, 53, 66, 83, 106;
%e A132427 106, 106, 129, 152, 181, 216, 259, 312, 378, 461, 567;
%e A132427 567, 567, 673, 779, 908, 1060, 1241, 1457, 1716, 2028, 2406, 2867, 3434; ...
%t A132427 t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, k-2]; t[n_, 0] := t[n, 0] = t[n-1, 2n-2]; t[n_, 1] := t[n, 0]; t[0, 0] = 1; Flatten[ Table[t[n, k], {n, 0, 7}, {k, 0, 2 n}]] (* _Jean-François Alcover_, Jun 18 2012 *)
%o A132427 (PARI) T(n,k)=local(A=[1]);if(2*n<k || k<0,0,if(n==0 && k==0,1,if(k==0 || k==1,T(n-1,2*n-2), for(i=1,n,A=Vec(Ser(concat([A[ #A],0],A))/(1-x)));A[k+1])))
%Y A132427 Cf. A125273; A132428 (central terms).
%K A132427 nonn,tabf
%O A132427 0,4
%A A132427 _Paul D. Hanna_, Aug 21 2007