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Search: id:A126265
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| A126265 |
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Triangle of coefficients of q in e.g.f. that satisfies: A(x,q) = exp( q*x*A(q*x,q) ), read by rows of [n*(n-1)/2 + 1] terms in row n for n>=0. |
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+0
7
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| 1, 1, 1, 2, 1, 6, 3, 6, 1, 12, 24, 28, 24, 12, 24, 1, 20, 90, 140, 245, 120, 240, 140, 120, 60, 120, 1, 30, 240, 660, 1320, 1626, 1920, 2100, 1560, 1830, 1440, 1440, 840, 720, 360, 720, 1, 42, 525, 2450, 6195, 12432, 15127, 23310, 21000, 26250, 19320, 26502, 19320
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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Row sums equal A000272(n) = (n+1)^(n-1). Last term in rows are the factorials. Coeffients of q in {[x^n] A(x,q)} when read backward converge to the sequence: [1,1/2,1,7/6,2,2,85/24,11/3,65/12,19/3,357/40,19/2,111/8,123/8,81/4,...].
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FORMULA
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G.f.: A(x,q) = Sum_{n>=0} (x^n/n!)*q^n*Sum_{k=0..n*(n-1)/2} T(n,k)*q^k.
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EXAMPLE
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G.f.: A(x,q) = 1 + q*x + (1 + 2*q)*q^2*x^2/2! +
(1 + 6*q + 3*q^2 + 6*q^3)*q^3*x^3/3! +
(1 + 12*q + 24*q^2 + 28*q^3 + 24*q^4 + 12*q^5 + 24*q^6)*q^4*x^4/4! +...
Triangle begins:
1;
1;
1, 2;
1, 6, 3, 6;
1, 12, 24, 28, 24, 12, 24;
1, 20, 90, 140, 245, 120, 240, 140, 120, 60, 120;
1, 30, 240, 660, 1320, 1626, 1920, 2100, 1560, 1830, 1440, 1440, 840, 720, 360, 720; ...
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PROGRAM
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(PARI) {T(n, k)=local(A=x); for(i=1, n, A=x*exp(subst(A, x, q*x+x*O(x^n)))); if(k>n*(n-1)/2|k<0, 0, Vec(Vec(A)[n+1]*n!+q*O(q^(n*(n+1)/2)))[k+1])}
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CROSSREFS
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Cf. A000272 (row sums); diagonals: A126266, A126267.
Sequence in context: A097288 A100014 A062566 this_sequence A124441 A026191 A050137
Adjacent sequences: A126262 A126263 A126264 this_sequence A126266 A126267 A126268
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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), Dec 22 2006
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