A205812
a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k).
Original entry on oeis.org
1, 11, 70, 719, 7806, 122534, 2097278, 43444159, 1000262653, 25997950846, 743008372734, 23312187863054, 793714773262334, 29197324076701078, 1152921975865606140, 48663045048486723199, 2185911559738696663038, 104128351926393946602653, 5242880000000000000524286
Offset: 1
L.g.f.: L(x) = x + 11*x^2/2 + 70*x^3/3 + 719*x^4/4 + 7806*x^5/5 +...
Exponentiation yields the g.f. of A205811:
exp(L(x)) = 1 + x + 6*x^2 + 29*x^3 + 221*x^4 + 1897*x^5 + 23502*x^6 +...
Illustration of terms.
a(2) = 2*sigma(2,1) + 1*sigma(2,2) = 2*3 + 1*5 = 11;
a(3) = 3*sigma(3,1) + 3*sigma(3,2) + 1*sigma(3,3) = 3*4 + 3*10 + 1*28 = 70;
a(4) = 4*sigma(4,1) + 6*sigma(4,2) + 4*sigma(4,3) + 1*sigma(4,3) = 4*7 + 6*21 + 4*73 + 1*273 = 719.
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Table[Sum[Binomial[n, k]*DivisorSigma[k, n], {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 08 2016 *)
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{a(n)=sum(k=1,n,binomial(n,k)*sigma(n,k))}
A205814
G.f.: Product_{n>=1} [ (1 - 2^n*x^n) / (1 - (n+2)^n*x^n) ]^(1/n).
Original entry on oeis.org
1, 1, 9, 54, 482, 4239, 55561, 785554, 14133055, 285547760, 6666380256, 172748192767, 4974178683908, 156462697434990, 5354832107694444, 197710292330150160, 7839473395324929677, 332071887435037103895, 14968498613432649146050, 715294449027151380463781
Offset: 0
G.f.: A(x) = 1 + x + 9*x^2 + 54*x^3 + 482*x^4 + 4239*x^5 + 55561*x^6 +...
where the g.f. equals the product:
A(x) = (1-2*x)/(1-3*x) * ((1-2^2*x^2)/(1-4^2*x^2))^(1/2) * ((1-2^3*x^3)/(1-5^3*x^3))^(1/3) * ((1-2^4*x^4)/(1-6^4*x^4))^(1/4) * ((1-2^5*x^5)/(1-7^5*x^5))^(1/5) *...
The logarithm equals the l.g.f. of A205815:
log(A(x)) = x + 17*x^2/2 + 136*x^3/3 + 1585*x^4/4 + 16986*x^5/5 +...
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{a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*sum(k=1, m, binomial(m, k)*sigma(m, k)*2^(m-k))+x*O(x^n))), n)}
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{a(n)=polcoeff(prod(k=1, n, ((1-2^k*x^k)/(1-(k+2)^k*x^k +x*O(x^n)))^(1/k)), n)}
A206766
a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k) * 3^(n-k).
Original entry on oeis.org
1, 23, 226, 3039, 33306, 594902, 10012010, 220553599, 5170061143, 138942811678, 4049569009674, 130045043225838, 4503599691290714, 168477832912220134, 6746676272050878036, 288487396687082933759, 13107200000016921588858, 630907565930072760920429
Offset: 1
L.g.f.: L(x) = x + 23*x^2/2 + 226*x^3/3 + 3039*x^4/4 + 33306*x^5/5 +...
Exponentiation yields the g.f. of A206765:
exp(L(x)) = 1 + x + 12*x^2 + 87*x^3 + 907*x^4 + 8393*x^5 + 118932*x^6 +...
Illustration of terms.
a(2) = 2*sigma(2,1)*3 + 1*sigma(2,2)*1 = 2*3*3 + 1*5*1 = 23;
a(3) = 3*sigma(3,1)*9 + 3*sigma(3,2)*3 + 1*sigma(3,3)*1 = 3*4*9 + 3*10*3 + 1*28*1 = 226;
a(4) = 4*sigma(4,1)*27 + 6*sigma(4,2)*9 + 4*sigma(4,3)*3 + 1*sigma(4,4)*1 = 4*7*27 + 6*21*9 + 4*73*3 + 1*273*1 = 3039.
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{a(n)=sum(k=1, n, binomial(n, k)*sigma(n, k)*3^(n-k))}
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{a(n)=n*polcoeff(sum(k=1, n, (1/k)*log((1-3^k*x^k)/(1-(k+3)^k*x^k +x*O(x^n)))), n)}
for(n=1,21,print1(a(n),", "))
A206764
a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k) * (-1)^(n-k).
Original entry on oeis.org
1, -1, 10, 79, 1026, 15686, 279938, 5771359, 134218243, 3487832974, 100000000002, 3138673052878, 106993205379074, 3937454749863382, 155568096631586820, 6568441588686506943, 295147905179352825858, 14063102470280932000757, 708235345355337676357634
Offset: 1
L.g.f.: L(x) = x - x^2/2 + 10*x^3/3 + 79*x^4/4 + 1026*x^5/5 + 15686*x^6/6 +...
Exponentiation yields the g.f. of A206763:
exp(L(x)) = 1 + x + 3*x^3 + 23*x^4 + 225*x^5 + 2824*x^6 + 42670*x^7 +...
Illustration of terms.
a(2) = -2*sigma(2,1) + 1*sigma(2,2) = -2*3 + 1*5 = -1;
a(3) = 3*sigma(3,1) - 3*sigma(3,2) + 1*sigma(3,3) = 3*4 - 3*10 + 1*28 = 10;
a(4) = -4*sigma(4,1) + 6*sigma(4,2) - 4*sigma(4,3) + 1*sigma(4,4) = -4*7 + 6*21 - 4*73 + 1*273 = 79.
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Table[Sum[Binomial[n, k] * DivisorSigma[k, n] * (-1)^(n-k), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 25 2024 *)
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{a(n)=sum(k=1, n, binomial(n, k)*sigma(n, k)*(-1)^(n-k))}
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{a(n)=n*polcoeff(sum(k=1, n, (1/k)*log((1-(-x)^k)/(1-(k-1)^k*x^k +x*O(x^n)))), n)}
for(n=1,21,print1(a(n),", "))
Showing 1-4 of 4 results.
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