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1, 3, 4, 7, 7, 9, 12, 11, 15, 14, 19, 19, 19, 22, 23, 27, 26, 27, 31, 30, 33, 35, 35, 40, 39, 39, 40, 47, 47, 47, 46, 51, 53, 52, 55, 55, 61, 60, 57, 67, 62, 69, 65, 64, 77, 71, 77, 72, 75, 83, 76, 81, 85, 85, 88, 85, 91, 92, 91, 95, 92, 103, 97, 99, 102, 105, 107, 104, 111
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internal format)
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = Sum_{m=n(n+1)/2..n(n+1)/2+n} [x^m] S(x)^2 for n>=0 where S(x) = Sum_{n>=0} x^(n(n+1)/2).
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EXAMPLE
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The coefficients in the square of the series:
S = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 +...
begin: [(1),(2,1),(2,2,0),(3,2,0,2),(2,2,1,2,0),(2,4,0,2,0,1),...];
the sums of the grouped coefficients yield the initial terms of this sequence.
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MATHEMATICA
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t[n_, k_] := Module[{s = Sum[x^(m*(m+1)/2), {m, 0, k+1}]+O[x]^((k+1)*(k+2)/2)}, Sum[Coefficient[s^n, x, m], {m, k*(k+1)/2, k*(k+1)/2+k}]]; Table[t[2, k], {k, 0, 68}] (* Jean-François Alcover, Nov 18 2013 *)
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PROG
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(PARI) {a(n)=local(S=sum(m=0, n+1, x^(m*(m+1)/2))+O(x^((n+1)*(n+2)/2))); sum(m=n*(n+1)/2, n*(n+1)/2+n, polcoeff(S^2, m))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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