A291698 -id:A291698 - cp's OEIS frontend

A124577 Define p(alpha) to be the number of H-conjugacy classes where H is a Young subgroup of type alpha of the symmetric group S_n. Then a(n) = sum p(alpha) where |alpha| = n and alpha has at most n parts.

1, 1, 6, 39, 356, 4055, 57786, 983535, 19520264, 441967518, 11235798510, 316719689506, 9800860032876, 330230585628437, 12032866998445818, 471416196117401340, 19758835313514076176, 882185444649249777913, 41797472220815112375966, 2094455101139881670407954

Offset: 0

Richard Bayley (r.t.bayley(AT)qmul.ac.uk), Nov 05 2006

nonn

p((0,n)) = A000041, p((1,n)) = A000070, p((2,n)) = A093695;
Also main diagonal of A209664. - Wouter Meeussen, Mar 11 2012
Number of partitions of n into n sorts of parts. a(2) = 6: [2a], [2b], [1a,1a], [1a,1b], [1b,1a], [1b,1b]. - Alois P. Heinz, Sep 08 2014

			E.g p((2,1)) = # H-conjugacy classes of S_3 where H = Yng((2,1)) isom S_2 times S_1 . Then a(3) = p((3)) + p((2,1)) + p((2,0,1)) + p((1,2)) + p((1,1,1))+ p((1,0,2)+ p((0,2,1)) + p((0,1,2)) + p((0,0,3)) = 3+4+4+4+6+4+3+4+4+3 = 39.

Alois P. Heinz, Table of n, a(n) for n = 0..250
Richard Bayley, Homepage.
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.
Wikipedia, Symmetric Polynomials

Cf. A124578, A000041, A000070, A093695, A209664-A209673, A291698.

Main diagonal of A246935.

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-> b(n$3):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 08 2014

p[n_Integer, v_] := Sum[Subscript[x, j]^n, {j, v}];
p[par_List, v_] := Times @@ (p[#, v] & /@ par);
Tr /@ Table[(p[#, l] & /@ IntegerPartitions[l]) /. Subscript[x, ] -> 1, {l, 19}] (* _Wouter Meeussen, Mar 11 2012 *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]]; a[n_] := b[n, n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

{a(n)=polcoeff(1/prod(k=1,n,1-n*x^k +x*O(x^n)),n)} \\ Paul D. Hanna, Nov 26 2009

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) we do the following. sum_alpha p(alpha)x^alpha = prod_(b in Phi) prod_(k = 1)^infinity 1/(1- x^(c(b) times k )) = prod_(b in Phi) prod_(k = 1)^infinity (1+ x^(k times C(b)) + x^(2k times C(b)) + x^(3k times C(b)) + ....)
From Paul D. Hanna, Nov 26 2009: (Start)
a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k) for n>0.
a(n) = Sum_{k=1..n} A008284(n,k)*n^k, where A008284(n,k) = number of partitions of n in which the greatest part is k, 1<=k<=n. (End)
a(n) ~ n^n * (1 + 1/n + 2/n^2 + 3/n^3 + 5/n^4 + 7/n^5 + 11/n^6 + 15/n^7 + 22/n^8 + 30/n^9 + 42/n^10), where the coefficients are A000041(k)/n^k. - Vaclav Kotesovec, Mar 19 2015

Extended with formula by Paul D. Hanna, Nov 26 2009

a(0) inserted and more terms from Alois P. Heinz, Sep 08 2014

A340103 a(n) = [x^n] Product_{k>=1} (1 + n^(k-1)*x^k).

Original entry on oeis.org

1, 1, 2, 12, 80, 875, 10584, 170471, 2949120, 63772920, 1441000000, 38818444632, 1089573617664, 35185728919614, 1175820172477440, 44425722744140625, 1722925924631969792, 74364737115532234518, 3291298649632850485248, 159785357022861166517580, 7932051456000000000000000

Offset: 0

Ilya Gutkovskiy, Apr 24 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..385

Cf. A000009, A003056, A008289, A291698, A292305, A304961, A338697, A344094.

Table[SeriesCoefficient[Product[(1 + n^(k - 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Unprotect[Power]; 0^0 = 1; Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] n^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 20}]
Join[{1}, Table[SeriesCoefficient[n*QPochhammer[-1/n, n*x]/(n+1), {x, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 09 2021 *)

a(n) = Sum_{k=0..A003056(n)} q(n,k) * n^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ c * n^(n-1), where c = BesselI(1,2) = A096789 = 1.590636854637329... - Vaclav Kotesovec, May 09 2021

A261569 Expansion of Product_{k>=1} (1 + 5*x^k).

Original entry on oeis.org

1, 5, 5, 30, 30, 55, 180, 205, 330, 480, 1230, 1380, 2255, 3030, 4530, 8555, 10680, 15330, 21330, 29730, 39480, 67380, 81505, 116280, 153030, 210930, 270805, 370080, 534330, 675480, 900480, 1180380, 1544130, 1997280, 2597280, 3304805, 4581180, 5653080

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

In general, for a fixed integer m > 0, if g.f. = Product_{k>=1} (1 + m*x^k) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m+1)*Pi)*n^(3/4)), where c = Pi^2/6 + log(m)^2/2 + polylog(2, -1/m) = -polylog(2, -m). - Vaclav Kotesovec, Jan 04 2016

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A032302, A032308, A261568, A291698.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 5*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015

nmax = 40; CoefficientList[Series[Product[1 + 5*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*5^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(QPochhammer[-5, x]/6 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(6*Pi)*n^(3/4)), where c = Pi^2/6 + log(5)^2/2 + polylog(2, -1/5) = 2.74927912606080829002558751537626864449... . - Vaclav Kotesovec, Jan 04 2016

G.f.: Sum_{i>=0} 5^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018

A286957 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + k*x^j).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 2, 0, 1, 4, 3, 6, 2, 0, 1, 5, 4, 12, 6, 3, 0, 1, 6, 5, 20, 12, 10, 4, 0, 1, 7, 6, 30, 20, 21, 18, 5, 0, 1, 8, 7, 42, 30, 36, 48, 22, 6, 0, 1, 9, 8, 56, 42, 55, 100, 57, 30, 8, 0, 1, 10, 9, 72, 56, 78, 180, 116, 84, 42, 10, 0, 1, 11, 10, 90, 72, 105, 294, 205, 180, 120, 66, 12, 0

Offset: 0

Ilya Gutkovskiy, May 17 2017

A(n,k) is the number of partitions of n into distinct parts of k sorts: the parts are unordered, but not the sorts.

			Square array begins:
1,  1,   1,   1,   1,   1,  ...
0,  1,   2,   3,   4,   5,  ...
0,  1,   2,   3,   4,   5,  ...
0,  2,   6,  12,  20,  30,  ...
0,  2,   6,  12,  20,  30,  ...
0,  3,  10,  21,  36,  55,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to partitions

Columns k=0-5 give: A000007, A000009, A032302, A032308, A261568, A261569.
Main diagonal gives A291698.
Cf. A246935.

Table[Function[k, SeriesCoefficient[Product[(1 + k x^i), {i, 1, Infinity}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-k, x]/(1 + k), {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 + k*x^j).

A292305 a(n) = [x^n] Product_{k>=1} (1 + n^k*x^k).

Original entry on oeis.org

1, 1, 4, 54, 512, 9375, 186624, 4117715, 100663296, 3099363912, 100000000000, 3423740047332, 133741506723840, 5451751918660554, 244464150162276352, 11823135040283203125, 590295810358705651712, 31435129951680797038726, 1809934771463640728469504

Offset: 0

Vaclav Kotesovec, Sep 14 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A265949, A292190, A291698, A292306.

nmax = 20; Table[SeriesCoefficient[Product[(1+n^k*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
Flatten[{1, Table[n^n*PartitionsQ[n], {n, 1, 20}]}]

a(n) = n^n * A000009(n).

a(n) ~ exp(Pi*sqrt(n/3)) * n^(n-3/4) / (4*3^(1/4)).

A292304 a(n) = [x^n] Product_{k>=1} (1 + n^2*x^k).

Original entry on oeis.org

1, 1, 4, 90, 272, 1275, 49284, 124901, 536640, 1620648, 104040100, 223290012, 880969104, 2485978170, 7454471332, 592164776475, 1138401673472, 4109108002310, 10877348160900, 30962024560494, 72270337440400, 7523649856001916, 13202150810778116, 44577985082575400

Offset: 0

Vaclav Kotesovec, Sep 14 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A092484, A291698.

nmax = 30; Table[SeriesCoefficient[Product[(1+n^2*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
Table[SeriesCoefficient[QPochhammer[-n^2, x, 1 + n]/(1 + n^2), {x, 0, n}], {n, 0, 30}]

Conjecture: a(n) ~ exp(2*sqrt((Pi^2/6 + 2*log(n)^2)*n)) * (Pi^2/6 + 2*log(n)^2)^(1/4) / (2 * sqrt(Pi) * n^(7/4)).

A298987 a(n) = [x^n] Product_{k>=1} (1 + n*x^k)^k.

Original entry on oeis.org

1, 1, 4, 27, 80, 400, 1908, 6223, 31296, 116478, 450100, 1828915, 7360848, 26906828, 95776772, 403908975, 1421758720, 5072014447, 18481180644, 68350964211, 246180936400, 827642046294, 2958748580084, 10294629775620, 36607347335232, 120800714172500, 407951731319860, 1405943613730899

Offset: 0

Ilya Gutkovskiy, Jan 31 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A026007, A261562, A266857, A266891, A291698, A297322, A298985, A298986, A298988.

Table[SeriesCoefficient[Product[(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

A292132 Main diagonal of A292131.

Original entry on oeis.org

1, -1, -2, 6, 12, 45, -150, -203, -840, -1872, 6390, 8580, 30084, 69108, 165802, -494565, -514320, -1997296, -4202298, -10175526, -18908420, 49930440, 54032770, 190735688, 405256872, 948210600, 1751280726, 3624555168, -7676955468, -6724059944, -26741354430

Offset: 0

Seiichi Manyama, Sep 09 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A124577, A291698, A292131.

{a(n) = polcoeff(prod(k=1, n, 1-n*x^k+x*O(x^n)), n)}

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

A292134 Main diagonal of A292133.

Original entry on oeis.org

1, -1, 2, -21, 220, -2705, 40926, -733537, 15124216, -352606050, 9174382490, -263533561852, 8283376452948, -282795708021411, 10420847619031710, -412243715452039440, 17425722339237083120, -783844576340696848341, 37384875796116662077194

Offset: 0

Seiichi Manyama, Sep 09 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A124577, A291698.

{a(n) = polcoeff(1/prod(k=1, n, 1+n*x^k+x*O(x^n)), n)}

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

a(n) ~ (-1)^n * n^n * (1 - 1/n + 2/n^2 - 3/n^3 + 5/n^4 - 7/n^5 + 11/n^6 - 15/n^7 + 22/n^8 - 30/n^9 + 42/n^10 - ...), for coefficients, see A000041. - Vaclav Kotesovec, Aug 21 2018

A303188 a(n) = [x^n] Product_{k=1..n} (1 + (n - k + 1)*x^k).

Original entry on oeis.org

1, 1, 1, 7, 9, 23, 148, 221, 526, 1040, 6767, 9664, 23456, 43943, 91363, 499028, 736410, 1650395, 3107540, 6210372, 10819270, 57864166, 80663444, 179915133, 324882691, 640398244, 1087149284, 2039724322, 9121580902, 12913282685, 27250167385, 48645989650, 92634730208, 156124357449

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

nonn

			a(0) = 1;
a(1) = [x^1] (1 + x) = 1;
a(2) = [x^2] (1 + 2*x)*(1 + x^2) = 1;
a(3) = [x^3] (1 + 3*x)*(1 + 2*x^2)*(1 + x^3) = 7;
a(4) = [x^4] (1 + 4*x)*(1 + 3*x^2)*(1 + 2*x^3)*(1 + x^4) = 9;
a(5) = [x^5] (1 + 5*x)*(1 + 4*x^2)*(1 + 3*x^3)*(1 + 2*x^4)*(1 + x^5) = 23, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} (1 + (n - k + 1)*x^k) begins:
n = 0: (1), 0,  0,   0,   0,   0,  ...
n = 1:  1, (1), 0,   0,   0,   0,  ...
n = 2:  1,  2, (1),  2,   0,   0   ...
n = 3:  1,  3,  2,  (7),  3,   2,  ...
n = 4:  1,  4,  3,  14,  (9), 10,  ...
n = 5:  1,  5,  4,  23,  17, (23), ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A022629, A206229, A291698, A303175, A303189, A303190.

Table[SeriesCoefficient[Product[(1 + (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 33}]

cp's OEIS Frontend

A124577 Define p(alpha) to be the number of H-conjugacy classes where H is a Young subgroup of type alpha of the symmetric group S_n. Then a(n) = sum p(alpha) where |alpha| = n and alpha has at most n parts.

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A340103 a(n) = [x^n] Product_{k>=1} (1 + n^(k-1)*x^k).

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A261569 Expansion of Product_{k>=1} (1 + 5*x^k).

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A286957 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + k*x^j).

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A292305 a(n) = [x^n] Product_{k>=1} (1 + n^k*x^k).

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A292304 a(n) = [x^n] Product_{k>=1} (1 + n^2*x^k).

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A298987 a(n) = [x^n] Product_{k>=1} (1 + n*x^k)^k.

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A292132 Main diagonal of A292131.

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