A333882
Expansion of e.g.f. exp(Sum_{k>=0} x^(5*k + 1) / (5*k + 1)!).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 464, 1399, 7801, 45410, 216581, 853218, 2896002, 11708734, 79817500, 615700986, 4012571831, 21538473686, 98707812691, 501634082800, 3983368886226, 37404203343457, 305886831698593, 2069143637726674, 11924094649669375
Offset: 0
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nmax = 29; CoefficientList[Series[Exp[Sum[x^(5 k + 1)/(5 k + 1)!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 5]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 29}]
nmax = 30; CoefficientList[Series[Exp[x*HypergeometricPFQ[{}, {2/5, 3/5, 4/5, 6/5}, x^5/3125]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)
A352428
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n,3*k+1) * a(n-3*k-1).
Original entry on oeis.org
1, 1, 2, 6, 25, 130, 810, 5881, 48806, 455706, 4727881, 53955682, 671730246, 9059714665, 131588822822, 2047796305470, 33992509701721, 599526848094850, 11195864285933682, 220692569175568729, 4579248276057441926, 99767702172338210898, 2277136869014579978473, 54336724559407913237122
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 3 k + 1] a[n - 3 k - 1], {k, 0, Floor[(n - 1)/3]}]; Table[a[n], {n, 0, 23}]
nmax = 23; CoefficientList[Series[1/(1 - Sum[x^(3 k + 1)/(3 k + 1)!, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
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my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\3, x^(3*k+1)/(3*k+1)!)))) \\ Seiichi Manyama, Mar 23 2022
A333883
Expansion of e.g.f. exp(Sum_{k>=0} x^(6*k + 1) / (6*k + 1)!).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1718, 5163, 32281, 217921, 1188709, 5291353, 20031170, 66744741, 267996541, 2030569465, 18368560519, 138812739409, 853152218102, 4409607501927, 19826125988257, 99717123889777, 871344991322017, 9658479225877057
Offset: 0
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nmax = 30; CoefficientList[Series[Exp[Sum[x^(6 k + 1)/(6 k + 1)!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 6]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Exp[x*HypergeometricPFQ[{}, {1/3, 1/2, 2/3, 5/6, 7/6}, x^6/46656]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)
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