A300512
Expansion of e.g.f. log(Sum_{k>=0} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).
Original entry on oeis.org
0, 1, 1, -1, -1, 6, -1, -77, 203, 1344, -10692, -15862, 579611, -1518768, -32884753, 283168220, 1550435633, -38615194078, 44538307279, 4920513118440, -39485852954288, -546206846420721, 11322395643617278, 23746787652752639, -2713550731461618505, 17064642256532964421
Offset: 0
E.g.f.: A(x) = x/1! + x^2/2! - x^3/3! - x^4/4! + 6*x^5/5! - x^6/6! - 77*x^7/7! + 203*x^8/8! + ...
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a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(j*a(j)*
binomial(n, j)*t(n-j), j=1..n-1)/n))(combinat[numbpart])
end:
seq(a(n), n=0..30); # Alois P. Heinz, Mar 07 2018
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nmax = 25; CoefficientList[Series[Log[Sum[PartitionsP[k] x^k/k!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = PartitionsP[n] - Sum[k Binomial[n, k] PartitionsP[n - k] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 25}]
A300514
Expansion of e.g.f. exp(Sum_{k>=1} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).
Original entry on oeis.org
1, 1, 2, 6, 20, 79, 358, 1791, 9854, 58958, 379716, 2617320, 19197327, 149099827, 1221390172, 10515829901, 94865603724, 894302028718, 8788782784778, 89848652800152, 953666248076772, 10491219933196228, 119429574273909421, 1404835599743325765, 17052591331677804136
Offset: 0
E.g.f.: A(x) = 1 + x/1! + 2*x^2/2! + 6*x^3/3! + 20*x^4/4! + 79*x^5/5! + 358*x^6/6! + 1791*x^7/7! + ...
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b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
`if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
a:= proc(n) option remember; `if`(n=0, 1, add(
a(n-j)*binomial(n-1, j-1)*b(j), j=1..n))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Mar 07 2018
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nmax = 24; CoefficientList[Series[Exp[Sum[PartitionsQ[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[PartitionsQ[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]
A324236
Expansion of e.g.f. 1 / (1 - Sum_{k>=1} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).
Original entry on oeis.org
1, 1, 4, 21, 149, 1317, 13985, 173209, 2451844, 39044784, 690862770, 13446615722, 285510978887, 6567419023617, 162686428939423, 4317885767971448, 122241788335870103, 3677030054440996775, 117111150680951037907, 3937135961534144480556, 139328182441566999124409
Offset: 0
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a:= proc(n) option remember; `if`(n=0, 1, add(a(n-k)*
binomial(n, k)*combinat[numbpart](k), k=1..n))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Sep 02 2019
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nmax = 20; CoefficientList[Series[1/(1 - Sum[PartitionsP[k] x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] PartitionsP[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
Showing 1-3 of 3 results.
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