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A100226 Triangle, read by rows, of the coefficients of [x^k] in G100225(x)^n such that the row sums are 3^n-1 for n>0, where G100225(x) is the g.f. of A100225.

%I A100226 #10 Aug 30 2024 21:40:31
%S A100226 1,1,1,1,2,5,1,3,9,13,1,4,14,28,33,1,5,20,50,85,81,1,6,27,80,171,246,
%T A100226 197,1,7,35,119,301,553,693,477,1,8,44,168,486,1064,1724,1912,1153,1,
%U A100226 9,54,228,738,1854,3600,5220,5193,2785,1,10,65,300,1070,3012,6730,11760
%N A100226 Triangle, read by rows, of the coefficients of [x^k] in G100225(x)^n such that the row sums are 3^n-1 for n>0, where G100225(x) is the g.f. of A100225.
%C A100226 Main diagonal forms A100227. Secondary diagonal is: T(n+1,n) = (n+1)*A001333(n), where A001333 is the numerators of continued fraction convergents to sqrt(2). More generally, if g.f. F(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]F(x)^n, then F(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](F(x)+z*x)^n for all z and F(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2; the triangle formed from powers of F(x) will have the g.f.: G(x,y)=(1-2*x*y+m*x^2*y^2)/((1-x*y)*(1-(m-1)*x*y-x^2*y^2-x*(1-x*y))).
%F A100226 G.f.: A(x, y)=(1-2*x*y+3*x^2*y^2)/((1-x*y)*(1-2*x*y-x^2*y^2-x*(1-x*y))).
%e A100226 Rows begin:
%e A100226   [1],
%e A100226   [1,1],
%e A100226   [1,2,5],
%e A100226   [1,3,9,13],
%e A100226   [1,4,14,28,33],
%e A100226   [1,5,20,50,85,81],
%e A100226   [1,6,27,80,171,246,197],
%e A100226   [1,7,35,119,301,553,693,477],
%e A100226   [1,8,44,168,486,1064,1724,1912,1153],...
%e A100226 where row sums form 3^n-1 for n>0:
%e A100226 3^1-1 = 1+1
%e A100226 3^2-1 = 1+2+5
%e A100226 3^3-1 = 1+3+9+13
%e A100226 3^4-1 = 1+4+14+28+33
%e A100226 3^5-1 = 1+5+20+50+85+81.
%e A100226 The main diagonal forms A100227 = [1,1,5,13,33,81,197,477,...], where Sum_{n>=1} (A100227(n)/n)*x^n = log((1-x)/(1-2*x-x^2)).
%o A100226 (PARI) T(n,k,m=3)=if(n<k || k<0,0,if(k==0,1, polcoeff(((1+(m-1)*x+sqrt(1+2*(m-3)*x+(m^2-2*m+5)*x^2+x*O(x^k)))/2)^n,k)))
%Y A100226 Cf. A100225, A100227, A001333.
%K A100226 nonn,tabl
%O A100226 0,5
%A A100226 _Paul D. Hanna_, Nov 28 2004