The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A296601 L.g.f.: -log(Product_{k>=1} (1 - k*x^k)^k) = Sum_{n>=1} a(n)*x^n/n. 3
 1, 9, 28, 81, 126, 330, 344, 833, 973, 1754, 1332, 5034, 2198, 5658, 8688, 13313, 4914, 28779, 6860, 54106, 45752, 33482, 12168, 254954, 93751, 78906, 255880, 505698, 24390, 1510700, 29792, 1671169, 1791312, 647114, 2819544, 12637371, 50654, 2282346, 14779520, 34058298, 68922, 68084220 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Seiichi Manyama, Table of n, a(n) for n = 1..5000 FORMULA G.f.: Sum_{k>=1} k^3*x^k/(1 - k*x^k). a(n) = Sum_{d|n} d^(n/d+2). a(p) = p^3 + 1 where p is a prime. From Seiichi Manyama, Jun 24 2019: (Start) Suppose given two sequences f(n) and g(n), n>0, we define a new sequence a(n), n>0, by a(n) = Sum_{d|n} d*f(d)*g(d)^(n/d). L.g.f.: -log(Product_{n>0} (1 - g(n)*x^n)^f(n)) = Sum_{n>0} a(n)*x^n/n. (See A266964.) If we set f(n) = n and g(n) = n, we get this sequence. (End) EXAMPLE L.g.f.: L(x) = x + 9*x^2/2 + 28*x^3/3 + 81*x^4/4 + 126*x^5/5 + 330*x^6/6 + 344*x^7/7 + 833*x^8/8 + 973*x^9/9 + ... exp(L(x)) = 1 + x + 5*x^2 + 14*x^3 + 42*x^4 + 103*x^5 + 289*x^6 + 690*x^7 + 1771*x^8 + 4206*x^9 + ... + A266941(n)*x^n + ... MATHEMATICA nmax = 42; Rest[CoefficientList[Series[-Log[Product[(1 - k x^k)^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]] nmax = 42; Rest[CoefficientList[Series[Sum[k^3 x^k/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x]] a[n_] := Sum[d^(n/d + 2), {d, Divisors[n]}]; Table[a[n], {n, 1, 42}] PROG (PARI) N=66; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k*x^k)^k)))) \\ Seiichi Manyama, Jun 02 2019 CROSSREFS Column k=2 of A308502. Cf. A001157, A078308, A266941, A266964. Sequence in context: A277065 A001158 A171215 * A294567 A053819 A294287 Adjacent sequences:  A296598 A296599 A296600 * A296602 A296603 A296604 KEYWORD nonn AUTHOR Ilya Gutkovskiy, May 20 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 21 16:54 EST 2020. Contains 331114 sequences. (Running on oeis4.)