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Search: id:A100229
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| A100229 |
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Triangle, read by rows, of the coefficients of [x^k] in G100228(x)^n such that the row sums are 4^n-1 for n>0, where G100228(x) is the g.f. of A100228. |
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4
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| 1, 1, 2, 1, 4, 10, 1, 6, 21, 35, 1, 8, 36, 92, 118, 1, 10, 55, 185, 380, 392, 1, 12, 78, 322, 879, 1506, 1297, 1, 14, 105, 511, 1715, 3948, 5803, 4286, 1, 16, 136, 760, 3004, 8536, 17020, 21904, 14158, 1, 18, 171, 1077, 4878, 16344, 40395, 71109, 81387, 46763
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OFFSET
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0,3
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COMMENT
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The main diagonal forms A100230. Secondary diagonal is T(n+1,n) = (n+1)*A052924(n). 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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FORMULA
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G.f.: A(x, y)=(1-2*x*y+4*x^2*y^2)/((1-x*y)*(1-3*x*y-x^2*y^2-x*(1-x*y))).
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EXAMPLE
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Rows begin:
[1],
[1,2],
[1,4,10],
[1,6,21,35],
[1,8,36,92,118],
[1,10,55,185,380,392],
[1,12,78,322,879,1506,1297],
[1,14,105,511,1715,3948,5803,4286],
[1,16,136,760,3004,8536,17020,21904,14158],...
where row sums form 4^n-1 for n>0:
4^1-1 = 1+2 = 3
4^2-1 = 1+4+10 = 15
4^3-1 = 1+6+21+35 = 63
4^4-1 = 1+8+36+92+118 = 255
4^5-1 = 1+10+55+185+380+392 = 1023.
The main diagonal forms A100230 = [1,2,10,35,118,392,1297,...],
where Sum_{n>=1} A100230(n)/n*x^n = log((1-x)/(1-3*x-x^2)).
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PROGRAM
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(PARI) {T(n, k, m=4)=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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Cf. A100228, A100230, A052924.
Sequence in context: A097949 A117338 A137634 this_sequence A071949 A156919 A038195
Adjacent sequences: A100226 A100227 A100228 this_sequence A100230 A100231 A100232
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Nov 29 2004
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