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Search: id:A127126
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| A127126 |
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Triangle, read by rows, where the g.f. of column k, C_k(x), is defined by the recurrence: C_k(x) = [ 1 + Sum_{n>=k+1} C_n(x)*x^(n-k) ]^(k+1) for k>=0. |
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+0
10
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| 1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 77, 54, 18, 4, 1, 587, 412, 139, 30, 5, 1, 5484, 3834, 1314, 284, 45, 6, 1, 60582, 42131, 14658, 3217, 505, 63, 7, 1, 771261, 533558, 188012, 42100, 6680, 818, 84, 8, 1, 11102828, 7645065, 2721462, 621936, 100621, 12387
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OFFSET
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0,4
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COMMENT
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This is a variant of triangles: A127082, A124328.
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EXAMPLE
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C_k = [ 1 + x*C_{k+1} + x^2*C_{k+2} + x^3*C_{k+3} +... ]^(k+1).
The columns are generated by working backwards:
C_3 = [ 1 + x*C_4 + x^2*C_5 + x^3*C_6 + x^4*C_7 +... ]^4;
C_2 = [ 1 + x*C_3 + x^2*C_4 + x^3*C_5 + x^4*C_6 +... ]^3;
C_1 = [ 1 + x*C_2 + x^2*C_3 + x^3*C_4 + x^4*C_5 +... ]^2;
C_0 = [ 1 + x*C_1 + x^2*C_2 + x^3*C_3 + x^4*C_4 +... ]^1.
The triangle begins:
1;
1, 1;
3, 2, 1;
13, 9, 3, 1;
77, 54, 18, 4, 1;
587, 412, 139, 30, 5, 1;
5484, 3834, 1314, 284, 45, 6, 1;
60582, 42131, 14658, 3217, 505, 63, 7, 1;
771261, 533558, 188012, 42100, 6680, 818, 84, 8, 1;
11102828, 7645065, 2721462, 621936, 100621, 12387, 1239, 108, 9, 1; ...
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PROGRAM
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(PARI) {T(n, k)=if(n==k, 1, polcoeff( (1 + x*sum(r=k+1, n, x^(r-k-1)*sum(c=k+1, r, T(r, c))) +x*O(x^n))^(k+1), n-k))}
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CROSSREFS
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Columns: A127127, A127128, A127129, A127130; central terms: A127134; variants: A127082, A124328.
Sequence in context: A132845 A129652 A154921 this_sequence A161133 A112911 A152405
Adjacent sequences: A127123 A127124 A127125 this_sequence A127127 A127128 A127129
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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), Jan 05 2007
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