cp's OEIS Frontend

A100232 Triangle, read by rows, of the coefficients of [x^k] in G100231(x)^n such that the row sums are 5^n-1 for n>0, where G100231(x) is the g.f. of A100231.

%I A100232 #7 Jun 13 2017 22:14:28
%S A100232 1,1,3,1,6,17,1,9,39,75,1,12,70,220,321,1,15,110,470,1165,1363,1,18,
%T A100232 159,852,2895,5922,5777,1,21,217,1393,5943,16807,29267,24475,1,24,284,
%U A100232 2120,10822,38536,93468,141688,103681,1,27,360,3060,18126,77274,236748
%N A100232 Triangle, read by rows, of the coefficients of [x^k] in G100231(x)^n such that the row sums are 5^n-1 for n>0, where G100231(x) is the g.f. of A100231.
%C A100232 The main diagonal forms A100233. Secondary diagonal is: T(n+1,n) = (n+1)*A033887(n) = (n+1)*Fibonacci(3*n+1). 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 A100232 G.f.: A(x, y)=(1-2*x*y+5*x^2*y^2)/((1-x*y)*(1-4*x*y-x^2*y^2-x*(1-x*y))).
%e A100232 Rows begin:
%e A100232 [1],
%e A100232 [1,3],
%e A100232 [1,6,17],
%e A100232 [1,9,39,75],
%e A100232 [1,12,70,220,321],
%e A100232 [1,15,110,470,1165,1363],
%e A100232 [1,18,159,852,2895,5922,5777],
%e A100232 [1,21,217,1393,5943,16807,29267,24475],
%e A100232 [1,24,284,2120,10822,38536,93468,141688,103681],...
%e A100232 where row sums form 5^n-1 for n>0:
%e A100232 5^1-1 = 1+3 = 4
%e A100232 5^2-1 = 1+6+17 = 24
%e A100232 5^3-1 = 1+9+39+75 = 124
%e A100232 5^4-1 = 1+12+70+220+321 = 624
%e A100232 5^5-1 = 1+15+110+470+1165+1363 = 3124.
%e A100232 The main diagonal forms A100233 = [1,3,17,75,321,1363,5777,...],
%e A100232 where Sum_{n>=1} A100233(n)/n*x^n = log((1-x)/(1-4*x-x^2)).
%o A100232 (PARI) T(n,k,m=5)=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 A100232 Cf. A100231, A100233, A033887, A100229.
%K A100232 nonn,tabl
%O A100232 0,3
%A A100232 _Paul D. Hanna_, Nov 29 2004