A323339
Numerator of the sum of inverse products of parts in all compositions of n.
Original entry on oeis.org
1, 1, 3, 7, 11, 347, 3289, 1011, 38371, 136553, 4320019, 12528587, 40771123, 29346499543, 129990006917, 1927874590951, 903657004321, 437445829053473, 12456509813711881, 187206004658210129, 1974369484466728177, 1967745662306280217, 21401375717067880189
Offset: 0
1/1, 1/1, 3/2, 7/3, 11/3, 347/60, 3289/360, 1011/70, 38371/1680, 136553/3780, 4320019/75600, 12528587/138600, 40771123/285120, ... = A323339/A323340
Cf.
A000142,
A007840,
A011782,
A088305,
A177208,
A177209,
A322364,
A322365,
A322380,
A322381,
A323290,
A323291.
-
b:= proc(n) option remember;
`if`(n=0, 1, add(b(n-j)/j, j=1..n))
end:
a:= n-> numer(b(n)):
seq(a(n), n=0..25);
-
nmax = 20; Numerator[CoefficientList[Series[1/(1 + Log[1-x]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Feb 12 2024 *)
A322364
Numerator of the sum of inverse products of parts in all partitions of n.
Original entry on oeis.org
1, 1, 3, 11, 7, 27, 581, 4583, 2327, 69761, 775643, 147941, 30601201, 30679433, 10928023, 6516099439, 445868889691, 298288331489, 7327135996801, 1029216937671847, 14361631943741, 837902013393451, 2766939485246012129, 274082602410356881, 835547516381094139939
Offset: 0
1/1, 1/1, 3/2, 11/6, 7/3, 27/10, 581/180, 4583/1260, 2327/560, 69761/15120, 775643/151200, 147941/26400, 30601201/4989600, 30679433/4633200 ... = A322364/A322365
- Alois P. Heinz, Table of n, a(n) for n = 0..505
- A. Knopfmacher, J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
- D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
- D. Zeilberger, N. Zeilberger, Fractional Counting of Integer Partitions, 2018.
Cf.
A000041,
A006906,
A080130,
A177208,
A177209,
A322380,
A322381,
A323290,
A323291,
A323339,
A323340.
-
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
b(n, i-1) +b(n-i, min(i, n-i))/i)
end:
a:= n-> numer(b(n$2)):
seq(a(n), n=0..30);
-
b[n_, i_] := b[n, i] = If[n==0||i==1, 1, b[n, i-1] + b[n-i, Min[i, n-i]]/i];
a[n_] := Numerator[b[n, n]];
a /@ Range[0, 30] (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)
-
a(n) = {my(s=0); forpart(p=n, s += 1/vecprod(Vec(p))); numerator(s);} \\ Michel Marcus, Apr 29 2020
A322380
Numerator of the sum of inverse products of parts in all strict partitions of n.
Original entry on oeis.org
1, 1, 1, 5, 7, 37, 79, 173, 101, 127, 1033, 1571, 200069, 2564519, 5126711, 25661369, 532393, 431100529, 1855391, 1533985991, 48977868113, 342880481117, 342289639579, 435979161889, 1308720597671, 373092965489, 7824703695283, 24141028973, 31250466692609
Offset: 0
1/1, 1/1, 1/2, 5/6, 7/12, 37/60, 79/120, 173/280, 101/168, 127/210, 1033/1680, 1571/2640, 200069/332640, 2564519/4324320, 5126711/8648640, ... = A322380/A322381
- Alois P. Heinz, Table of n, a(n) for n = 0..1268
- Andreas B. G. Blobel, An Asymptotic Form of the Generating Function Prod_{k=1,oo} (1+x^k/k), arXiv:1904.07808 [math.CO], 2019.
- Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 137.
- A. Knopfmacher and J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
- D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
- D. Zeilberger, N. Zeilberger, Fractional Counting of Integer Partitions, 2018.
Cf.
A000009,
A000142,
A007838,
A022629,
A080130,
A177208,
A177209,
A322364,
A322365,
A323290,
A323291,
A323339,
A323340.
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +b(n-i, min(i-1, n-i))/i))
end:
a:= n-> numer(b(n$2)):
seq(a(n), n=0..30);
-
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[i - 1, n - i]]/i]];
a[n_] := Numerator[b[n, n]];
a /@ Range[0, 30] (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)
A323291
Denominator of the sum of inverse products of cycle sizes in all permutations of [n].
Original entry on oeis.org
1, 1, 2, 6, 12, 20, 360, 2520, 1680, 15120, 151200, 554400, 9979200, 129729600, 605404800, 27243216000, 54486432000, 308756448000, 5557616064000, 8122669632000, 351982350720000, 22174888095360000, 25676186215680000, 3740164458750720000, 67322960257512960000
Offset: 0
-
b:= proc(n) option remember; `if`(n=0, 1, add(
b(n-j)*binomial(n-1, j-1)*(j-1)!/j, j=1..n))
end:
a:= n-> denom(b(n)):
seq(a(n), n=0..25);
-
nmax = 30; Denominator[CoefficientList[Series[Exp[PolyLog[2, x]], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Feb 12 2024 *)
Showing 1-4 of 4 results.
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