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A160571
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G.f.: Product_{n>=1} (1 + x^n + x^(n+1)).
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0
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1, 1, 2, 3, 5, 7, 10, 15, 21, 28, 38, 52, 70, 92, 119, 154, 200, 258, 329, 416, 523, 655, 819, 1022, 1269, 1566, 1924, 2357, 2879, 3507, 4263, 5170, 6250, 7530, 9048, 10849, 12980, 15496, 18466, 21967, 26079, 30894, 36526, 43109, 50792, 59743, 70160
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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FORMULA
| G.f.: A(x) = Sum_{n>=0} x^(n*(3*n+1)/2)*(1+x)^n*(1 + x^(2*n+1)*(1+x)) * Product_{k=1..n} (1 + x^k*(1+x))/(1-x^k) due to Sylvester's identity.
G.f.: A(x) = Sum_{n>=0} x^(n*(n+1)/2)*(1+x)^n / Product_{k=1..n} (1-x^k).
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EXAMPLE
| G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 10*x^6 + 15*x^7 +...
G.f.: A(x) = (1+x*(1+x))*(1+x^2*(1+x))*(1+x^3*(1+x))*(1+x^4*(1+x))*...
G.f.: A(x) = (1+x*(1+x)) + x^2*(1+x)*(1 + x^3*(1+x))*(1+x*(1+x))/(1-x) + x^7*(1+x)^2*(1 + x^5*(1+x))*(1+x*(1+x))*(1+x^2*(1+x))/((1-x)*(1-x^2)) + x^15*(1+x)^3*(1 + x^7*(1+x))*(1+x*(1+x))*(1+x^2*(1+x))*(1+x^3*(1+x))/((1-x)*(1-x^2)*(1-x^3)) +...
G.f.: A(x) = 1 + x*(1+x)/(1-x) + x^3*(1+x)^2/((1-x)*(1-x^2)) + x^6*(1+x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^10*(1+x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) +...
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PROG
| (PARI) a(n)=polcoeff(prod(k=1, n, 1+x^k*(1+x) +x*O(x^n)), n)
(PARI) {a(n)=local(A=1+x); A=sum(m=0, n, x^(m*(3*m+1)/2)*(1+x)^m*(1 + x^(2*m+1)*A)*prod(k=1, m, (1+A*x^k)/(1-x^k+x*O(x^n)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); A=sum(m=0, n, x^(m*(m+1)/2)*(1+x)^m/prod(k=1, m, 1-x^k +x*O(x^n))); polcoeff(A, n)}
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CROSSREFS
| Cf. A151552.
Sequence in context: A092021 A022475 A011972 * A184641 A076972 A170877
Adjacent sequences: A160568 A160569 A160570 * A160572 A160573 A160574
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), May 20 2009, May 21 2009, Jul 17 2011
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