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A100226
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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.
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3
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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, 197, 1, 7, 35, 119, 301, 553, 693, 477, 1, 8, 44, 168, 486, 1064, 1724, 1912, 1153, 1, 9, 54, 228, 738, 1854, 3600, 5220, 5193, 2785, 1, 10, 65, 300, 1070, 3012, 6730, 11760
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OFFSET
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0,5
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COMMENTS
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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))).
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LINKS
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FORMULA
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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))).
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EXAMPLE
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Rows begin:
[1],
[1,1],
[1,2,5],
[1,3,9,13],
[1,4,14,28,33],
[1,5,20,50,85,81],
[1,6,27,80,171,246,197],
[1,7,35,119,301,553,693,477],
[1,8,44,168,486,1064,1724,1912,1153],...
where row sums form 3^n-1 for n>0:
3^1-1 = 1+1
3^2-1 = 1+2+5
3^3-1 = 1+3+9+13
3^4-1 = 1+4+14+28+33
3^5-1 = 1+5+20+50+85+81.
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).
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PROG
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(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)))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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