A061160
Numerators in expansion of Euler transform of b(n) = 1/3.
Original entry on oeis.org
1, 1, 5, 50, 215, 646, 8711, 25475, 105925, 3091270, 11691247, 36809705, 445872155, 1364113925, 5085042010, 50975292560, 183383680088, 588817265695, 19512559194875, 62369303509475, 224877933068647, 2214198452392027, 7686538660149565, 25124342108522750
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Geoffrey B. Campbell, Some n-space q-binomial theorem extensions and similar identities, arXiv:1906.07526 [math.NT], 2019.
- Geoffrey B. Campbell, Continued Fractions for partition generating functions, arXiv:2301.12945 [math.CO], 2023.
- Geoffrey B. Campbell, Vector Partition Identities for 2D, 3D and nD Lattices, arXiv:2302.01091 [math.CO], 2023.
- Geoffrey B. Campbell and A. Zujev, Some almost partition theoretic identities, Preprint, 2016.
- N. J. A. Sloane, Transforms
-
b:= proc(n) option remember; `if`(n=0, 1, add(add(
d/3, d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
end:
a:= n-> numer(b(n)):
seq(a(n), n=0..30); # Alois P. Heinz, Jul 28 2017
-
c[n_] := c[n] = If[n == 0, 1,
(1/(3n)) Sum[c[n-k] DivisorSigma[1, k], {k, 1, n}]];
a[n_] := Numerator[c[n]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 24 2022 *)
A061161
Numerators in expansion of Euler transform of b(n) = 1/4.
Original entry on oeis.org
1, 1, 13, 55, 1235, 4615, 55801, 200343, 8977475, 36804235, 367235363, 1444888289, 32062742231, 120729974115, 1205864254225, 5201022002071, 395884671433315, 1603069490974835, 15989295873680415, 64312573140322525, 1250332447587844829, 5262481040435242585
Offset: 0
-
b:= proc(n) option remember; `if`(n=0, 1, add(add(
d/4, d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
end:
a:= n-> numer(b(n)):
seq(a(n), n=0..30); # Alois P. Heinz, Jul 28 2017
-
c[n_] := c[n] = If[n == 0, 1,
(1/(4n)) Sum[c[n-k] DivisorSigma[1, k], {k, 1, n}]];
a[n_] := Numerator[c[n]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 24 2022 *)
A343204
Numerators of coefficients in expansion of Product_{k>=1} (1 + x^k)^(1/2).
Original entry on oeis.org
1, 1, 3, 13, 67, 239, 1031, 2501, 36579, 109915, 468653, 1043851, 9395751, 21232827, 97493519, 235880373, 7717800611, 17385733651, 82456426833, 175398844079, 1578297716013, 3634938193489, 15867173716609, 34517119775523, 619312307079687, 1363237700933583
Offset: 0
1, 1/2, 3/8, 13/16, 67/128, 239/256, 1031/1024, 2501/2048, 36579/32768, 109915/65536, 468653/262144, 1043851/524288, ...
-
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
`if`(d::odd, d/2, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
a:= n-> numer(b(n)):
seq(a(n), n=0..25); # Alois P. Heinz, Apr 12 2021
-
nmax = 25; CoefficientList[Series[Product[(1 + x^k)^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator
a[n_] := a[n] = If[n == 0, 1, (1/(2 n)) Sum[Sum[Mod[d, 2] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}] // Numerator
A282030
Numerators in an "almost partition identity".
Original entry on oeis.org
1, 1, 19, 343, 11305, 58349, 3230255, 15652637, 1076842043, 55727585675, 827556357229, 4462381098539, 472784126720165, 2370070888522337, 36233114985867725, 676572663482064349, 76896411800028508843, 428831636633480067995, 60264676461386071468577
Offset: 0
Showing 1-4 of 4 results.
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