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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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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, 1239, 108, 9, 1
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OFFSET
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0,4
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COMMENTS
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This is a variant of triangles: A127082, A124328.
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LINKS
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G. C. Greubel, Rows n = 0..50 of triangle, flattened
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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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MATHEMATICA
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T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k-1)*Sum[T[r, c], {c, k+1, r}], {r, k+1, n}] +x^(n+1))^(k+1), x, n-k]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 27 2020 *)
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PROG
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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 * A161133 A112911 A152405
Adjacent sequences: A127123 A127124 A127125 * 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, Jan 05 2007
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STATUS
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approved
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