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A318638
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Expansion of Sum_{n>=1} ( (3 + x^n)^n - 3^n ).
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5
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1, 6, 27, 109, 405, 1467, 5103, 17550, 59050, 197100, 649539, 2126991, 6908733, 22325625, 71744625, 229602925, 731794257, 2324602206, 7360989291, 23245524600, 73222475256, 230128853031, 721764371007, 2259440202825, 7060738412026, 22029517662984, 68630377426119, 213516777941712, 663426981193869, 2058911488612863, 6382625094934119, 19765549255048254, 61149666233193318
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = Sum_{d|n} 3^(d - n/d) * binomial(d, n/d). - Seiichi Manyama, Apr 24 2021
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EXAMPLE
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G.f.: A(x) = x + 6*x^2 + 27*x^3 + 109*x^4 + 405*x^5 + 1467*x^6 + 5103*x^7 + 17550*x^8 + 59050*x^9 + 197100*x^10 + 649539*x^11 + 2126991*x^12 + ...
such that
A(x) = x + (3 + x^2)^2 - 3^2 + (3 + x^3)^3 - 3^3 + (3 + x^4)^4 - 3^4 + (3 + x^5)^5 - 3^5 + (3 + x^6)^6 - 3^6 + (3 + x^7)^7 - 3^7 + ...
RELATED SERIES.
The g.f. A(x) equals following series at y = 3:
Sum_{n>=1} ((y + x^n)^n - y^n) = x + 2*y*x^2 + 3*y^2*x^3 + (4*y^3 + 1)*x^4 + 5*y^4*x^5 + (6*y^5 + 3*y)*x^6 + 7*y^6*x^7 + (8*y^7 + 6*y^2)*x^8 + (9*y^8 + 1)*x^9 + (10*y^9 + 10*y^3)*x^10 + 11*y^10*x^11 + (12*y^11 + 15*y^4 + 4*y)*x^12 + 13*y^12*x^13 + (14*y^13 + 21*y^5)*x^14 + (15*y^14 + 10*y^2)*x^15 + (16*y^15 + 28*y^6 + 1)*x^16 + ...
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PROG
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(PARI) {a(n) = polcoeff( sum(m=1, n, (x^m + 3 +x*O(x^n))^m - 3^m), n)}
for(n=1, 100, print1(a(n), ", "))
(PARI) a(n) = sumdiv(n, d, 3^(d-n/d)* binomial(d, n/d)); \\ Seiichi Manyama, Apr 24 2021
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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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