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A243919
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G.f.: Sum_{n>=0} (x^n - 1)^n * x^n / (1-x)^(n+1).
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6
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1, 0, 1, 2, 1, -2, -3, 1, 9, 13, 6, -13, -33, -39, -18, 33, 102, 159, 158, 46, -199, -525, -781, -744, -221, 808, 2089, 3101, 3212, 1933, -831, -4650, -8635, -11669, -12652, -10615, -4635, 6314, 23181, 45855, 71241, 91000, 90501, 51044, -43113, -193400, -374830, -529580, -573509
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OFFSET
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0,4
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COMMENTS
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This sequence has a complex pattern of signs.
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LINKS
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FORMULA
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G.f.: Sum_{n>=0} x^(n*(n+1)) / (1-x + x^(n+1))^(n+1).
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^k - 1)^k.
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1 - x^k)^(n-k) * x^(k^2).
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EXAMPLE
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G.f.: A(x) = 1 + x^2 + 2*x^3 + x^4 - 2*x^5 - 3*x^6 + x^7 + 9*x^8 + 13*x^9 +...
where we have the series identity:
A(x) = 1/(1-x) + (x-1)*x/(1-x)^2 + (x^2-1)^2*x^2/(1-x)^3 + (x^3-1)^3*x^3/(1-x)^4 + (x^4-1)^4*x^4/(1-x)^5 +...+ (x^n-1)^n * x^n / (1-x)^(n+1) +...
A(x) = 1 + x^2/(1-x+x^2)^2 + x^6/(1-x+x^3)^3 + x^12/(1-x+x^4)^4 + x^20/(1-x+x^5)^5 + x^30/(1-x+x^6)^6 +...+ x^(n*(n+1)) / (1-x + x^(n+1))^(n+1) +...
as well as the binomial identity:
A(x) = 1 + x*(1 + (x-1)) + x^2*(1 + 2*(x-1) + (x^2-1)^2) + x^3*(1 + 3*(x-1) + 3*(x^2-1)^2 + (x^3-1)^3) + x^4*(1 + 4*(x-1) + 6*(x^2-1)^2 + 4*(x^3-1)^3 + (x^4-1)^4) + x^5*(1 + 5*(x-1) + 10*(x^2-1)^2 + 10*(x^3-1)^3 + 5*(x^4-1)^4 + (x^5-1)^5) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (x^k-1)^k +...
A(x) = 1 + x*(0 + x) + x^2*(0 + 2*(1-x)*x + x^4) + x^3*(0 + 3*(1-x)^2*x + 3*(1-x^2)*x^4 + x^9) + x^4*(0 + 4*(1-x)^3*x + 6*(1-x^2)^2*x^4 + 4*(1-x^3)*x^9 + x^16) + x^5*(0 + 5*(1-x)^4*x + 10*(1-x^2)^3*x^4 + 10*(1-x^3)^2*x^9 + 5*(1-x^4)*x^16 + x^25) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (1-x^k)^(n-k) * x^(k^2) +...
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PROG
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(PARI) {a(n)=local(A); A=sum(m=0, n, (x^m - 1)^m * x^m / (1-x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))
(PARI) {a(n)=local(A); A=sum(m=0, sqrtint(n+1), x^(m*(m+1)) / (1-x + x^(m+1) +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)*(x^k-1)^k) +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)*(1-x^k)^(m-k)*x^(k^2)) +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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