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A100224
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.
2
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, 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, 10, 10, 45, 70, 135, 190, 230, 210, 122
OFFSET
0,6
COMMENTS
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+...
FORMULA
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))).
EXAMPLE
Rows begin:
[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],
[1,0,7,7,21,28,35,28],
[1,0,8,8,28,40,60,64,46],...
where row sums form 2^n-1 for n>0:
2^1-1 = 1+0 = 1
2^2-1 = 1+0+2 = 3
2^3-1 = 1+0+3+3 = 7
2^4-1 = 1+0+4+4+6 = 15
2^5-1 = 1+0+5+5+10+10 = 31.
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)).
PROG
(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)))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Nov 28 2004
STATUS
approved