A100224 - OEIS

%I #9 Aug 30 2024 21:40:44

%S 1,1,0,1,0,2,1,0,3,3,1,0,4,4,6,1,0,5,5,10,10,1,0,6,6,15,18,17,1,0,7,7,

%T 21,28,35,28,1,0,8,8,28,40,60,64,46,1,0,9,9,36,54,93,117,117,75,1,0,

%U 10,10,45,70,135,190,230,210,122

%N Triangle, read by rows, of the coefficients of [x^k] in G100224(x)^n such that the row sums are 2^n-1 for n>0, where G100224(x) is the g.f. of A100224.

%C Diagonals are: T(n,n)=A001610(n-1) for n>0, with T(0,0)=1, T(n+1,n)=A006490(n), T(n+2,n)=A006491(n), T(n+3,n)=A006492(n), T(n+4,n)=A006493(n). The ratio of the generating functions of any two adjacent diagonals gives: (1-x)/(1-x-x^2) = 1+ x^2+ x^3+ 2*x^4+ 3*x^5+ 5*x^6+ 8*x^7+ 13*x^8+...

%F G.f.: A(x, y)=(1-2*x*y+2*x^2*y^2)/((1-x*y)*(1-x*y-x^2*y^2-x*(1-x*y))).

%e Rows begin:

%e   [1],

%e   [1,0],

%e   [1,0,2],

%e   [1,0,3,3],

%e   [1,0,4,4,6],

%e   [1,0,5,5,10,10],

%e   [1,0,6,6,15,18,17],

%e   [1,0,7,7,21,28,35,28],

%e   [1,0,8,8,28,40,60,64,46],...

%e where row sums form 2^n-1 for n>0:

%e 2^1-1 = 1+0 = 1

%e 2^2-1 = 1+0+2 = 3

%e 2^3-1 = 1+0+3+3 = 7

%e 2^4-1 = 1+0+4+4+6 = 15

%e 2^5-1 = 1+0+5+5+10+10 = 31.

%e The main diagonal forms A001610 = [0,2,3,6,10,17,...], where Sum_{n>=1} (A001610(n-1)/n)*x^n = log((1-x)/(1-x-x^2)).

%o (PARI) T(n,k)=if(n<k || k<0,0,if(k==0,1, polcoeff(((1+x+sqrt(1-2*x+5*x^2+x*O(x^k)))/2)^n,k)))

%Y Cf. A100223, A001610, A006490, A006491, A006492, A006493.

%K nonn,tabl

%O 0,6

%A _Paul D. Hanna_, Nov 28 2004