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A325997
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G.f.: Sum_{n>=0} (n+1) * (x + x^n)^n.
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4
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1, 4, 3, 10, 8, 18, 7, 40, 9, 44, 41, 54, 13, 150, 15, 88, 127, 168, 19, 298, 21, 324, 275, 180, 25, 854, 132, 238, 524, 774, 31, 1286, 33, 1180, 893, 378, 674, 2998, 39, 460, 1406, 3744, 43, 2790, 45, 3458, 4397, 648, 49, 8420, 303, 4714, 2960, 6270, 55, 7060, 6492, 9120, 4049, 990, 61, 30748, 63, 1120, 7697, 13788, 15082, 17626, 69, 16834, 6971, 28788, 73, 48088, 75, 1558, 39792, 25578, 12091, 41578, 81, 77874, 11540, 1890, 85, 121650, 58227
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OFFSET
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0,2
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COMMENTS
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More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
for any fixed integer k; here, k = 2 and p = x, q = x, r = 1.
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LINKS
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FORMULA
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G.f.: Sum_{n>=0} (n+1) * (x + x^n)^n.
G.f.: Sum_{n>=0} (n+1) * x^(n^2) / (1 - x^(n+1))^(n+2).
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EXAMPLE
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G.f.: A(x) = 1 + 4*x + 3*x^2 + 10*x^3 + 8*x^4 + 18*x^5 + 7*x^6 + 40*x^7 + 9*x^8 + 44*x^9 + 41*x^10 + 54*x^11 + 13*x^12 + 150*x^13 + 15*x^14 + 88*x^15 +...
where
A(x) = 1 + 2*(x + x) + 3*(x + x^2)^2 + 4*(x + x^3)^3 + 5*(x + x^4)^4 + 6*(x + x^5)^5 + 7*(x + x^6)^6 + 8*(x + x^7)^7 + 9*(x + x^8)^8 + 10*(x + x^9)^9 + ...
Also
A(x) = 1/(1-x)^2 + 2*x/(1 - x^2)^3 + 3*x^4/(1 - x^3)^4 + 4*x^9/(1 - x^4)^5 + 5*x^16/(1 - x^5)^6 + 6*x^25/(1 - x^6)^7 + 7*x^36/(1 - x^7)^8 + 8*x^49/(1 - x^8)^9 + 9*x^64/(1 - x^9)^10 + 10*x^81/(1 - x^10)^11 + ...
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MAPLE
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G:= add((n+1)*(x+x^n)^n, n=0..100):
S:= series(G, x, 101):
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PROG
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(PARI) {a(n)=polcoeff(sum(m=0, n, (m+1) * (x + x^m +x*O(x^n))^m), n)}
for(n=0, 100, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(m=0, n, (m+1) * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+2)), n)}
for(n=0, 100, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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