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Search: id:A127082
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| A127082 |
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Triangle, read by rows, where the g.f. of column k, C_k(x), is defined by the recursion: C_k(x) = [ 1 + Sum_{n>=1} x^n*C_{n-1+k}(x) ]^(k+1). |
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+0
11
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| 1, 1, 1, 2, 2, 1, 5, 7, 3, 1, 16, 28, 15, 4, 1, 64, 127, 85, 26, 5, 1, 308, 650, 531, 192, 40, 6, 1, 1728, 3737, 3600, 1551, 365, 57, 7, 1, 11046, 23996, 26266, 13416, 3635, 620, 77, 8, 1, 79065, 170866, 205353, 122770, 38556, 7356, 973, 100, 9, 1, 625049, 1338578
(list; table; graph; listen)
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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 triangle A124328.
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EXAMPLE
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C_k = [ 1 + x*C_k + x^2*C_{k+1} + x^3*C_{k+2} +... ]^(k+1).
The columns are generated by working backwards:
C_3 = [ 1 + x*C_3 + x^2*C_4 + x^3*C_5 + x^4*C_6 +... ]^4;
C_2 = [ 1 + x*C_2 + x^2*C_3 + x^3*C_4 + x^4*C_5 +... ]^3;
C_1 = [ 1 + x*C_1 + x^2*C_2 + x^3*C_3 + x^4*C_4 +... ]^2;
C_0 = [ 1 + x*C_0 + x^2*C_1 + x^3*C_2 + x^4*C_3 +... ]^1;
thus the row sums equal column 0 shift left.
The triangle begins:
1;
1, 1;
2, 2, 1;
5, 7, 3, 1;
16, 28, 15, 4, 1;
64, 127, 85, 26, 5, 1;
308, 650, 531, 192, 40, 6, 1;
1728, 3737, 3600, 1551, 365, 57, 7, 1;
11046, 23996, 26266, 13416, 3635, 620, 77, 8, 1;
79065, 170866, 205353, 122770, 38556, 7356, 973, 100, 9, 1;
625049, 1338578, 1716582, 1180496, 429515, 92730, 13412, 1440,126,10,1;
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PROGRAM
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(PARI) {T(n, k)=if(n==k, 1, polcoeff( (1 + x*sum(r=k, n-1, x^(r-k)*sum(c=k, r, T(r, c)))+x*O(x^n))^(k+1), n-k))}
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CROSSREFS
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Cf. variant: A124328; columns: A127083, A127084, A127085, A127086; A127090 (central terms).
Sequence in context: A056857 A129100 A162382 this_sequence A065052 A128308 A109152
Adjacent sequences: A127079 A127080 A127081 this_sequence A127083 A127084 A127085
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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 04 2007
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