A333981
a(0) = 0; a(n) = 2^(n-1) + (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * 2^(k-1) * (n-k) * a(n-k).
Original entry on oeis.org
0, 1, 4, 34, 576, 16296, 691408, 41069568, 3252707328, 331218217600, 42159307194624, 6558777387076608, 1224428872399488000, 270143735036619436032, 69534931015726331203584, 20651854796028308275851264, 7009822878720340562163007488, 2696576146784893519040303235072, 1166999997199470676471689819258880
Offset: 0
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a[0] = 0; a[n_] := a[n] = 2^(n - 1) + (1/n) Sum[Binomial[n, k]^2 2^(k - 1) (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[-Log[(3 - BesselI[0, 2 Sqrt[2 x]])/2], {x, 0, nmax}], x] Range[0, nmax]!^2
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@CachedFunction
def a(n): return 0 if (n==0) else 2^(n-1) + (1/n)*sum(binomial(n,k)^2 *2^(k-1)*(n-k)*a(n-k) for k in (1..n-1)) # a= A333981
[a(n) for n in (0..30)] # G. C. Greubel, Jun 09 2022
A333982
a(0) = 0; a(n) = 3^(n-1) + (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * 3^(k-1) * (n-k) * a(n-k).
Original entry on oeis.org
0, 1, 5, 48, 909, 28836, 1371384, 91308708, 8106024861, 925225277004, 132007041682380, 23019553116101268, 4817014157800460664, 1191268407723761654964, 343706793228408937835772, 114423311913128119741898268, 43534429651349601213257298621, 18771927426013054800534345817884, 9106204442628918977341144456510260
Offset: 0
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a[0] = 0; a[n_] := a[n] = 3^(n - 1) + (1/n) Sum[Binomial[n, k]^2 3^(k - 1) (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n],{n, 0, 18}]
nmax = 18; CoefficientList[Series[-Log[(4 - BesselI[0, 2 Sqrt[3 x]])/3], {x, 0, nmax}], x] Range[0, nmax]!^2
A333983
a(0) = 0; a(n) = 4^(n-1) + (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * 4^(k-1) * (n-k) * a(n-k).
Original entry on oeis.org
0, 1, 6, 64, 1328, 46336, 2423040, 177379840, 17314109440, 2172895068160, 340868882825216, 65356107645583360, 15037174515952517120, 4088810357694136320000, 1297103066111891262668800, 474788193071044243776077824, 198617395218460028950533898240, 94165608216423156721014443868160
Offset: 0
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a[0] = 0; a[n_] := a[n] = 4^(n - 1) + (1/n) Sum[Binomial[n, k]^2 4^(k - 1) (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n],{n, 0, 17}]
nmax = 17; CoefficientList[Series[-Log[(5 - BesselI[0, 4 Sqrt[x]])/4], {x, 0, nmax}], x] Range[0, nmax]!^2
A333984
a(0) = 0; a(n) = 5^(n-1) + (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * 5^(k-1) * (n-k) * a(n-k).
Original entry on oeis.org
0, 1, 7, 82, 1839, 69630, 3950650, 313747050, 33224570175, 4523562983350, 769859662962750, 160137417877796250, 39971947204607486250, 11791483690935887486250, 4058152793413483423916250, 1611522009185095020022068750, 731368135285580087866788609375, 376178084508304435598172207843750
Offset: 0
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a[0] = 0; a[n_] := a[n] = 5^(n - 1) + (1/n) Sum[Binomial[n, k]^2 5^(k - 1) (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n],{n, 0, 17}]
nmax = 17; CoefficientList[Series[-Log[(6 - BesselI[0, 2 Sqrt[5 x]])/5], {x, 0, nmax}], x] Range[0, nmax]!^2
Showing 1-4 of 4 results.