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A318638 Expansion of Sum_{n>=1} ( (3 + x^n)^n - 3^n ).

%I A318638 #20 Oct 30 2023 09:51:56
%S A318638 1,6,27,109,405,1467,5103,17550,59050,197100,649539,2126991,6908733,
%T A318638 22325625,71744625,229602925,731794257,2324602206,7360989291,
%U A318638 23245524600,73222475256,230128853031,721764371007,2259440202825,7060738412026,22029517662984,68630377426119,213516777941712,663426981193869,2058911488612863,6382625094934119,19765549255048254,61149666233193318
%N A318638 Expansion of Sum_{n>=1} ( (3 + x^n)^n - 3^n ).
%H A318638 Paul D. Hanna, <a href="/A318638/b318638.txt">Table of n, a(n) for n = 1..1024</a>
%F A318638 a(n) ~ n * 3^(n-1). - _Vaclav Kotesovec_, Oct 10 2020
%F A318638 a(n) = Sum_{d|n} 3^(d - n/d) * binomial(d, n/d). - _Seiichi Manyama_, Apr 24 2021
%F A318638 G.f.: Sum_{k >=1} x^(k^2)/(1-3*x^k)^(k+1). - _Seiichi Manyama_, Oct 30 2023
%e A318638 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 + ...
%e A318638 such that
%e A318638 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 + ...
%e A318638 RELATED SERIES.
%e A318638 The g.f. A(x) equals following series at y = 3:
%e A318638 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 + ...
%o A318638 (PARI) {a(n) = polcoeff( sum(m=1,n, (x^m + 3 +x*O(x^n))^m - 3^m), n)}
%o A318638 for(n=1,100, print1(a(n),", "))
%o A318638 (PARI) a(n) = sumdiv(n, d, 3^(d-n/d)* binomial(d, n/d)); \\ _Seiichi Manyama_, Apr 24 2021
%Y A318638 Cf. A318636, A318637, A338693.
%K A318638 nonn
%O A318638 1,2
%A A318638 _Paul D. Hanna_, Sep 07 2018