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A111607 Fourth column of A109626.

%I A111607 #8 Jan 29 2025 17:30:53
%S A111607 1,2,3,3,5,3,7,2,9,10,11,9,13,7,15,4,17,18,19,15,21,11,23,6,25,26,27,
%T A111607 21,29,15,31,8,33,34,35,27,37,19,39,10,41,42,43,33,45,23,47,12,49,50,
%U A111607 51,39,53,27,55,14,57,58,59,45,61,31,63,16,65,66,67,51,69,35,71,18,73,74,75
%N A111607 Fourth column of A109626.
%H A111607 G. C. Greubel, <a href="/A111607/b111607.txt">Table of n, a(n) for n = 1..5000</a>
%F A111607 G.f.: x*(1 + 2*x + 3*x^2 + 3*x^3 + 5*x^4 + 3*x^5 + 7*x^6 + 2*x^7 + 7*x^8 + 6*x^9 + 5*x^10 + 3*x^11 + 3*x^12 + x^13 + x^14)/(1-x^8)^2.
%t A111607 (* First program *)
%t A111607 f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 128}]]; g[n_, m_] := f[n][[m]]; Table[g[n, 4 + 1], {n, 75}]
%t A111607 (* Second program *)
%t A111607 CoefficientList[Series[(1+2*x+3*x^2+3*x^3+5*x^4+3*x^5+7*x^6+2*x^7+7*x^8 +6*x^9+5*x^10+3*x^11+3*x^12+x^13+x^14)/(1-x^8)^2, {x,0,100}], x] (* _G. C. Greubel_, Jan 29 2025 *)
%o A111607 (Magma)
%o A111607 R<x>:=PowerSeriesRing(Integers(), 102);
%o A111607 p:= func< x | x*(1+2*x+3*x^2+3*x^3 +5*x^4 +3*x^5 +7*x^6 +2*x^7 +7*x^8 +6*x^9 +5*x^10 +3*x^11 +3*x^12 +x^13 +x^14)/(1-x^8)^2 >;
%o A111607 Coefficients(R!( p(x) )); // _G. C. Greubel_, Jan 29 2025
%o A111607 (SageMath)
%o A111607 def p(x): return x*(1+2*x+3*x^2+3*x^3 +5*x^4 +3*x^5 +7*x^6 +2*x^7 +7*x^8 +6*x^9 +5*x^10 +3*x^11 +3*x^12 +x^13 +x^14)
%o A111607 def A111607_list(prec):
%o A111607     P.<x> = PowerSeriesRing(ZZ, prec)
%o A111607     return P( p(x)/(1-x^8)^2 ).list()
%o A111607 a=A111607_list(101); a[1:] # _G. C. Greubel_, Jan 29 2025
%Y A111607 Cf. A109626.
%K A111607 nonn
%O A111607 1,2
%A A111607 _Paul D. Hanna_ and _Robert G. Wilson v_, Aug 01 2005