A346936
a(n) = Sum_{d|n} mu(n/d) * binomial(5*d,d) / (4*d+1).
Original entry on oeis.org
1, 4, 34, 280, 2529, 23712, 231879, 2330160, 23950320, 250540836, 2658968129, 28558319744, 309831575759, 3390416555996, 37377257156716, 414741861215840, 4628362722856424, 51912988232308104, 584909606696793884, 6617078646710069720, 75134301594081157746, 855968478539048248916
Offset: 1
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Table[Sum[MoebiusMu[n/d] Binomial[5 d, d]/(4 d + 1), {d, Divisors[n]}], {n, 22}]
A346937
a(n) = Sum_{d|n} mu(n/d) * binomial(6*d,d) / (5*d+1).
Original entry on oeis.org
1, 5, 50, 500, 5480, 62776, 749397, 9203128, 115607259, 1478308780, 19180049927, 251857056364, 3340843549854, 44700484300317, 602574657421585, 8175951649914160, 111572030260242089, 1530312970224714489, 21085148778264281864, 291705220703240850760, 4050527291832419432577
Offset: 1
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Table[Sum[MoebiusMu[n/d] Binomial[6 d, d]/(5 d + 1), {d, Divisors[n]}], {n, 21}]
A346938
a(n) = Sum_{d|n} mu(n/d) * binomial(7*d,d) / (6*d+1).
Original entry on oeis.org
1, 6, 69, 812, 10471, 141702, 1997687, 28988856, 430321563, 6503342378, 99726673129, 1547847703500, 24269405074739, 383846166714410, 6116574500850339, 98106248277869040, 1582638261961640246, 25661404527359789034, 417980115131315136399, 6836064539918615002932
Offset: 1
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Table[Sum[MoebiusMu[n/d] Binomial[7 d, d]/(6 d + 1), {d, Divisors[n]}], {n, 20}]
A346939
a(n) = Sum_{d|n} mu(n/d) * binomial(8*d,d) / (7*d+1).
Original entry on oeis.org
1, 7, 91, 1232, 18277, 285285, 4638347, 77650784, 1329890613, 23190011435, 410333440535, 7349042707872, 132969010888279, 2426870701777445, 44627576949345735, 826044435331747776, 15378186970730687399, 287756293702214647875, 5409093674555090316299, 102094541350713952736608
Offset: 1
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Table[Sum[MoebiusMu[n/d] Binomial[8 d, d]/(7 d + 1), {d, Divisors[n]}], {n, 20}]
A380552
G.f. A(x) satisfies x = Sum_{n>=1} A( x^n*(1-x)^(3*n) ).
Original entry on oeis.org
1, 2, 14, 88, 611, 4372, 32889, 254384, 2017341, 16300550, 133767542, 1111727456, 9338434699, 79155402978, 676196048434, 5815796615520, 50318860986107, 437662918037250, 3824609516638443, 33563127916092808, 295655735395364616, 2613391671434553220, 23173063762591336049, 206066197523415007168
Offset: 1
G.f.: A(x) = x + 2*x^2 + 14*x^3 + 88*x^4 + 611*x^5 + 4372*x^6 + 32889*x^7 + 254384*x^8 + 2017341*x^9 + 16300550*x^10 + ...
where x = Sum_{n>=1} A( x^n*(1-x)^(3*n) ).
RELATED SERIES.
Sum_{n>=1} a(n) * x^n/(1-x^n) = x + 3*x^2 + 15*x^3 + 91*x^4 + 612*x^5 + 4389*x^6 + 32890*x^7 + 254475*x^8 + ... + A006632(n)*x^(n) + ...
which equals x*F(x)^3 where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.
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\\ As the Moebius transform of A006632 \\
{a(n) = sumdiv(n,d, moebius(n/d) * binomial(4*d-1,d-1)*3/(4*d-1) )}
for(n=1,30,print1(a(n),", "))
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\\ By definition x = Sum_{n>=1} A( x^n*(1-x)^(3*n) ) \\
{a(n) = my(V=[0,1]); for(i=0,n, V = concat(V,0); A = Ser(V);
V[#V] = polcoef(x - sum(m=1,#V, subst(A,x, x^m*(1-x)^(3*m) +x*O(x^#V)) ),#V-1)); V[n+1]}
for(n=1,30,print1(a(n),", "))
Showing 1-5 of 5 results.
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