A124577 -id:A124577 - cp's OEIS frontend

A292419 a(n) = [x^n] Product_{k>=1} (1 + nx^k) / (1 - nx^k).

1, 2, 12, 96, 872, 9960, 138180, 2298016, 44686224, 995739498, 24993249820, 697309946784, 21396151468536, 715827315312200, 25926440773118340, 1010478298772398080, 42162515927954808352, 1875027040759682964144, 88527520717734462201756, 4422273966757678408594560

Offset: 0

Vaclav Kotesovec, Sep 16 2017

nonn

Convolution of A291698 and A124577.

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A124577, A291698, A292418.

nmax = 25; Table[SeriesCoefficient[Product[(1+n*x^k)/(1-n*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]

{a(n)= polcoef(prod(k=1, n, ((1+n*x^k)/(1-n*x^k) +x*O(x^n))), n)};
for(n=0,20, print1(a(n), ", ")) \\ G. C. Greubel, Feb 02 2019

a(n) ~ 2 * n^n * (1 + 2/n + 4/n^2 + 8/n^3 + 14/n^4 + 24/n^5 + 40/n^6 + 64/n^7 + 100/n^8 + 154/n^9 + 232/n^10), for coefficients see A015128.

A124578 Define p(alpha,2) to be the number of H-conjugacy classes where H is an infant subgroup ( similar to Young subgroups of S_n) of type alpha of the hyperoctahedral group B_n. Then a(n) = sum p(alpha,2) where |alpha| = n and alpha has at most n parts.

Original entry on oeis.org

2, 16, 150, 1784, 25460

Offset: 1

Richard Bayley (r.t.bayley(AT)qmul.ac.uk), Nov 12 2006

p((n,0),2) = A000712. B_n can also be thought of as the signed permutation group. B_3 acts on the alphabet {1,2,3,bar{1}, bar{2}, bar{3}}. An infant subgroup of type (2,1) will be the subgroup which stabilizes the sets {1,bar{1}, 2, bar{2}} and {3,bar{3}}.

			E.g p((2,1),2) = # H-conjugacy classes of B_3 where H = Inft((2,1)) isom B_2 times B_1 . Then a(3) = p((3),2) + p((2,1),2) + p((2,0,1),2) + p((1,2),2) + p((1,1,1),2)+ p((1,0,2),2)+ p((0,3),2) + p((0,2,1),2) + p((0,1,2),2) + p((0,0,3),2) =10 + 16 + 16 + 16 + 24 + 16 + 10 + 16 + 16 +10 = 150

Richard Bayley, Relative Character Theory and the Hyperoctahedral Group, Ph.D. thesis, Queen Mary College, University of London, to be published 2007.
Steve Donkin, Invariant functions on Matrices, Math. Proc. Camb. Phil. Soc. 113 (1993) 23-43.

Richard Bayley, Homepage.

Cf. A124577, A000712.

Let x = x_1x_2x_3... and x^alpha = x_1^(alpha_1)x_2^(alpha_2)x_3^(alpha_3).... Let Phi = set of all primitive necklaces. If b is a primitive necklace then C(b) = Content(b) = (beta_1, beta_2,beta_3,.....) where beta_i = the number of times i occurs in b. For example if b=[11233] then C(b) = (2,1,2). To generate the p(alpha,2) we do the following. sum_alpha p(alpha,2)x^alpha = prod_(b in Phi) prod_(k = 1)^infinity 1/(1- x^(C(b) times k ))^2 = prod_(b in Phi) prod_(k = 1)^infinity (1+ x^(k times C(b)) + x^(2k times C(b)) + x^(3k times C(b)) + ....)^2

A303914 a(n) = [x^n] (1/(1 - x))Product_{k>=1} 1/(1 - nx^k).

Original entry on oeis.org

1, 2, 9, 55, 465, 5051, 69265, 1147287, 22307905, 497211049, 12484203601, 348391613615, 10691846920081, 357749800027465, 12958472141161457, 505088781523073326, 21076091000708067585, 937322034938743608556, 44256147057318887809993, 2210813717869831566759857, 116492226446226314836976401

Offset: 0

Ilya Gutkovskiy, May 02 2018

nonn

Index entries for sequences related to partitions

Cf. A000070, A124577, A246935, A303070.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
a:= n-> add(b(j$2, n), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 02 2018

Table[SeriesCoefficient[1/(1 - x) Product[1/(1 - n x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/(1 - x) Exp[Sum[n^k x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 20}]

a(n) = [x^n] (1/(1 - x))*exp(Sum_{k>=1} n^k*x^k/(k*(1 - x^k))).
a(n) = Sum_{j=0..n} A246935(j,n).
a(n) ~ n^n. - Vaclav Kotesovec, May 04 2018

A302836 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^(k^2)).

Original entry on oeis.org

1, 1, 4, 27, 260, 3150, 46872, 825944, 16810048, 387952668, 10010010100, 285526191874, 8921263237056, 303013028232642, 11116057874586840, 438023675344410000, 18451248777413066768, 827408674110381669305, 39353155876513869320412, 1978708139249503877752798

Offset: 0

Ilya Gutkovskiy, Apr 13 2018

nonn

Cf. A001156, A124577, A285245, A291583, A292417, A300974.

Table[SeriesCoefficient[Product[1/(1 - n x^k^2), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

a(n) ~ n^n * (1 + 1/n^3 + 1/n^6 + 1/n^8 + 1/n^9 + 1/n^11 + 1/n^12 + 1/n^14 + 2/n^15 + 1/n^16 + 1/n^17 + 2/n^18 + 1/n^19 + 1/n^20 + 2/n^21 + 1/n^22 + 2/n^23 + 4/n^24 + 1/n^25 + 2/n^26 + 4/n^27 + 1/n^28 + 2/n^29 + 5/n^30 + ...), for coefficients see A111178. - Vaclav Kotesovec, Apr 13 2018

A302837 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^prime(k)).

Original entry on oeis.org

1, 0, 2, 3, 16, 30, 252, 399, 4672, 8100, 111200, 193006, 3279888, 5631093, 114137072, 196030350, 4600308224, 7873377297, 210745840824, 359490499457, 10808589776800, 18389567969010, 613522956193532, 1041400845949218, 38179982045681856, 64675343605547500, 2584515863960831060

Offset: 0

Ilya Gutkovskiy, Apr 13 2018

nonn

Index entries for sequences related to partitions

Cf. A000607, A002098, A124577, A145519, A301971.

Table[SeriesCoefficient[Product[1/(1 - n x^Prime[k]), {k, 1, n}], {x, 0, n}], {n, 0, 26}]

cp's OEIS Frontend

A292419 a(n) = [x^n] Product_{k>=1} (1 + n*x^k) / (1 - n*x^k).

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A124578 Define p(alpha,2) to be the number of H-conjugacy classes where H is an infant subgroup ( similar to Young subgroups of S_n) of type alpha of the hyperoctahedral group B_n. Then a(n) = sum p(alpha,2) where |alpha| = n and alpha has at most n parts.

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A303914 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} 1/(1 - n*x^k).

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A302836 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^(k^2)).

Views

Author

Keywords

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Programs

Mathematica

Formula

A302837 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^prime(k)).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

A292419 a(n) = [x^n] Product_{k>=1} (1 + nx^k) / (1 - nx^k).

A303914 a(n) = [x^n] (1/(1 - x))Product_{k>=1} 1/(1 - nx^k).