A346935
a(n) = Sum_{d|n} mu(n/d) * binomial(4*d,d) / (3*d+1).
Original entry on oeis.org
1, 3, 21, 136, 968, 7059, 53819, 420592, 3362238, 27342916, 225568797, 1882926144, 15875338989, 134993712777, 1156393242330, 9969937070688, 86445222719723, 753310719641286, 6594154339031799, 57956002304003096, 511238042454487704, 4524678117713613419, 40166643855158315819
Offset: 1
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Table[Sum[MoebiusMu[n/d] Binomial[4 d, d]/(3 d + 1), {d, Divisors[n]}], {n, 23}]
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}]
A380553
G.f. A(x) satisfies x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).
Original entry on oeis.org
1, 3, 25, 200, 1770, 16351, 158223, 1577328, 16112031, 167708890, 1772645419, 18974340640, 205263418940, 2240623110285, 24648785800540, 272994642782048, 3041495503591364, 34064252952038769, 383302465665133013, 4331178750570145160, 49126274119206904221, 559128033687856289017
Offset: 1
G.f.: A(x) = x + 3*x^2 + 25*x^3 + 200*x^4 + 1770*x^5 + 16351*x^6 + 158223*x^7 + 1577328*x^8 + 16112031*x^9 + 167708890*x^10 + ...
where x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).
RELATED SERIES.
Sum_{n>=1} a(n) * x^n/(1-x^n) = x + 4*x^2 + 26*x^3 + 204*x^4 + 1771*x^5 + 16380*x^6 + 158224*x^7 + 1577532*x^8 + ... + A118971(n)*x^(n) + ...
which equals x*F(x)^4 where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.
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\\ As the Moebius transform of A118971 \\
{a(n) = sumdiv(n,d, moebius(n/d) * binomial(5*d-1,d-1)*4/(5*d-1) )}
for(n=1,30,print1(a(n),", "))
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\\ By definition x = Sum_{n>=1} A( x^n*(1-x)^(4*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)^(4*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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