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))}
A163191
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n,k)*sigma(n,k) for n>0 with a(0)=1.
Original entry on oeis.org
1, 0, 1, 8, 82, 1024, 15690, 279936, 5771363, 134218240, 3487832978, 100000000000, 3138673052884, 106993205379072, 3937454749863386, 155568096631586816, 6568441588686506948, 295147905179352825856, 14063102470280932000763, 708235345355337676357632
Offset: 0
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a[0] = 1; a[n_] := DivisorSum[n, (#-1)^n &]; Array[a, 20, 0] (* Amiram Eldar, Aug 15 2023 *)
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{a(n)=if(n==0,1,sumdiv(n,d,(d-1)^n))}
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{a(n)=if(n==0,1,sum(k=0,n,(-1)^(n-k)*binomial(n,k)*sigma(n,k)))}
A367509
a(n) = Sum_{d|n} (2*d+1)^n.
Original entry on oeis.org
3, 34, 370, 7267, 161294, 4960812, 170861562, 7019201348, 322728071069, 16705828227476, 952809758091074, 59628239376008854, 4052555153020570590, 297587425607933152700, 23465266173840431204452, 1978033864507607364591749, 177482997121587371955312038
Offset: 1
A367510
a(n) = Sum_{d|n} (n*d+1)^n.
Original entry on oeis.org
2, 34, 1064, 90707, 11889152, 2617716748, 781252097152, 320058240313028, 167630129865661440, 110581271670766466804, 89116503268963605948416, 86403662577880534613564934, 99045780329060163714773254144
Offset: 1
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Table[Total[(n*Divisors[n]+1)^n],{n,20}] (* Harvey P. Dale, Feb 06 2024 *)
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a(n) = sumdiv(n, d, (n*d+1)^n);
A367506
a(n) = Sum_{d|n} (d+n)^n.
Original entry on oeis.org
2, 25, 280, 6017, 107776, 3897218, 107510656, 4867995713, 204519070720, 10904505179450, 585061309782016, 38168392129581810, 2481946587976990720, 185404290282527361386, 14389574562121084305408, 1221867855128546542385409, 108430221517525671050739712
Offset: 1
A367507
a(n) = Sum_{d|n} (d+2)^n.
Original entry on oeis.org
3, 25, 152, 1633, 17050, 282594, 4785156, 101751713, 2359920499, 62200947098, 1792160571184, 56765070083650, 1946195069953698, 72080471103601322, 2862427829603252768, 121449533922042173249, 5480386857784931326102, 262149577935595805876315
Offset: 1
A367551
a(n) = Sum_{d|n} (d^2+1)^n.
Original entry on oeis.org
2, 29, 1008, 84162, 11881408, 2566742098, 781250000128, 318651789038947, 167619551409708544, 110462353708225871026, 89116503268220597579776, 86380568889558343409300388, 99045780329059370000000008192
Offset: 1
Showing 1-7 of 7 results.
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