A308509
Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n/d).
Original entry on oeis.org
1, 1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 9, 10, 9, 2, 1, 17, 28, 33, 6, 4, 1, 33, 82, 129, 26, 24, 2, 1, 65, 244, 513, 126, 182, 8, 4, 1, 129, 730, 2049, 626, 1458, 50, 41, 3, 1, 257, 2188, 8193, 3126, 11954, 344, 577, 37, 4, 1, 513, 6562, 32769, 15626, 99594, 2402, 8705, 811, 68, 2
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
2, 3, 5, 9, 17, 33, 65, ...
2, 4, 10, 28, 82, 244, 730, ...
3, 9, 33, 129, 513, 2049, 8193, ...
2, 6, 26, 126, 626, 3126, 15626, ...
4, 24, 182, 1458, 11954, 99594, 840242, ...
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T[n_, k_] := DivisorSum[n, #^(k*n/#) &]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)
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T(n,k) = sumdiv(n, d, (n/d)^(k*d));
matrix(9, 9, n, k, T(n,k-1)) \\ Michel Marcus, Jun 02 2019
A308696
a(n) = Sum_{d|n} d^(2*d).
Original entry on oeis.org
1, 17, 730, 65553, 9765626, 2176783082, 678223072850, 281474976776209, 150094635296999851, 100000000000009765642, 81402749386839761113322, 79496847203390846310290154, 91733330193268616658399616010, 123476695691247935826908004929122
Offset: 1
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a[n_] := DivisorSum[n, #^(2*#) &]; Array[a, 14] (* Amiram Eldar, May 09 2021 *)
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{a(n) = sumdiv(n, d, d^(2*d))}
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N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(2*k-1)))))
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N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(2*k)*x^k/(1-x^k)))
A151954
Expansion of Product_{k>0} (1-k^2*x^k)^(-1/k).
Original entry on oeis.org
1, 1, 3, 6, 16, 27, 79, 126, 331, 632, 1436, 2509, 6800, 11218, 26044, 51958, 114941, 205183, 502228, 875545, 2027193, 3963938, 8389190, 15504996, 37555290, 66502859, 145809046, 292860564, 621638120, 1156065731, 2701045579
Offset: 0
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nmax = 40; CoefficientList[Series[Product[(1-k^2*x^k)^(-1/k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 05 2017 *)
A308570
a(n) = sigma_{2*n}(n).
Original entry on oeis.org
1, 17, 730, 65793, 9765626, 2177317874, 678223072850, 281479271743489, 150094635684419611, 100000095367432689202, 81402749386839761113322, 79496851942053939878082786, 91733330193268616658399616010, 123476696151234472370970011268514
Offset: 1
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Table[DivisorSigma[2 n, n], {n, 1, 20}] (* Vaclav Kotesovec, Jun 08 2019 *)
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{a(n) = sigma(n, 2*n)}
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N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-(k^2*x)^k)^(1/k)))))
A073706
a(n) = Sum_{ d divides n } (n/d)^(3d).
Original entry on oeis.org
1, 9, 28, 129, 126, 1458, 344, 8705, 20413, 49394, 1332, 1104114, 2198, 2217546, 16305408, 33820673, 4914, 532253187, 6860, 2392632274, 10500716072, 8591716802, 12168, 422182489826, 30517593751, 549760658274, 7625984925160
Offset: 1
a(10) = (10/1)^(3*1) +(10/2)^(3*2) +(10/5)^(3*5) +(10/10)^(3*10) = 49394 because positive divisors of 10 are 1, 2, 5, 10.
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Table[Total[Quotient[n, x = Divisors[n]]^(3*x)], {n, 27}] (* Jayanta Basu, Jul 08 2013 *)
A294620
Expansion of Product_{k>0} (1 - k^2*x^k)^(1/k).
Original entry on oeis.org
1, -1, -2, -1, -3, 7, -12, 32, -10, -15, 77, 187, -760, 846, 1382, -4197, 1371, 6650, -9991, 19220, -32439, -80889, 290596, 127853, -1372003, 913414, 3253746, -6728692, 2302327, 14461937, -46087740, 66588519, 15702643, -357119564, 752905288, 310992687
Offset: 0
Showing 1-6 of 6 results.