A242587 -id:A242587 - cp's OEIS frontend

A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 14, 5, 0, 1, 5, 20, 39, 34, 7, 0, 1, 6, 30, 84, 129, 74, 11, 0, 1, 7, 42, 155, 356, 399, 166, 15, 0, 1, 8, 56, 258, 805, 1444, 1245, 350, 22, 0, 1, 9, 72, 399, 1590, 4055, 5876, 3783, 746, 30, 0

Offset: 0

Alois P. Heinz, Sep 08 2014

In general, column k > 1 is asymptotic to c * k^n, where c = Product_{j>=1} 1/(1-1/k^j) = 1/QPochhammer[1/k,1/k]. - Vaclav Kotesovec, Mar 19 2015

When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field of size k. - Geoffrey Critzer, Nov 11 2022

			A(2,2) = 6: [2a], [2b], [1a,1a], [1a,1b], [1b,1a], [1b,1b].
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,      1,      1, ...
  0,  1,   2,    3,     4,      5,      6,      7, ...
  0,  2,   6,   12,    20,     30,     42,     56, ...
  0,  3,  14,   39,    84,    155,    258,    399, ...
  0,  5,  34,  129,   356,    805,   1590,   2849, ...
  0,  7,  74,  399,  1444,   4055,   9582,  19999, ...
  0, 11, 166, 1245,  5876,  20455,  57786, 140441, ...
  0, 15, 350, 3783, 23604, 102455, 347010, 983535, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Columns k=0-10 give: A000007, A000041, A070933, A242587, A246936, A246937, A246938, A246939, A246940, A246941, A246942.
Rows n=0-4 give: A000012, A001477, A002378, A027444, A186636.
Main diagonal gives A124577.
Cf. A144064, A255970, A256130.

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, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

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_, k_] := b[n, n, k];  Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 03 2015, after Alois P. Heinz *)

G.f. of column k: Product_{i>=1} 1/(1-k*x^i).

T(n,k) = Sum_{i=0..k} C(k,i) * A255970(n,i).

A261582 Expansion of Product_{k>=1} 1/(1 + 3*x^k).

Original entry on oeis.org

1, -3, 6, -21, 69, -201, 591, -1785, 5406, -16194, 48426, -145380, 436641, -1309611, 3927399, -11783280, 35354139, -106059387, 318165729, -954506190, 2863556475, -8590643832, 25771817454, -77315531169, 231946940175, -695840583126, 2087520715788, -6262562872614

Offset: 0

Vaclav Kotesovec, Aug 25 2015

sign

Seiichi Manyama, Table of n, a(n) for n = 0..2094

Cf. A032308, A242587, A246935, A261565, A261567.

nmax = 40; CoefficientList[Series[Product[1/(1 + 3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*3^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x]
(O[x]^30 + 4/QPochhammer[-3, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) ~ c * (-3)^n, where c = Product_{j>=1} 1/(1-1/(-3)^j) = 1/QPochhammer[-1/3,-1/3] = 0.8212554466473167689981660621182786378...

G.f.: Sum_{i>=0} (-3)^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 13 2018

A292128 Expansion of Product_{k>=1} (1 - 3*x^k).

Original entry on oeis.org

1, -3, -3, 6, 6, 15, -12, -3, -30, -48, 6, -12, 15, 78, 186, -21, 168, 42, 42, -246, -408, -156, -399, -552, -498, -246, 213, 1248, -318, 1608, 1392, 2508, 1482, 2976, -480, -1011, 1500, -1704, -4296, -4206, -8499, -8652, -7626, -7050, -192, -13008, -480, -2118

Offset: 0

Seiichi Manyama, Sep 09 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Column k=3 of A292131.
Product_{k>=1} (1 - m*x^k): A010815 (m=1), A070877 (m=2), this sequence (m=3).
Cf. A032308, A242587, A292130.

N=66; x='x+O('x^N); Vec(prod(n=1, N, 1-3*x^n))

G.f.: Sum_{i>=0} (-3)^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 13 2018

A261565 Expansion of Product_{k>=1} (1/(1 - 3*x^k))^k.

Original entry on oeis.org

1, 3, 15, 54, 201, 672, 2268, 7266, 23208, 72414, 224652, 688929, 2103975, 6386907, 19337091, 58367817, 175905741, 529331190, 1591515297, 4781575074, 14359673454, 43108645230, 129387584991, 388283978589, 1165099808574, 3495782937135, 10488322595625

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

In general, for z > 1 or z < -1, if g.f. = Product_{k>=1} (1/(1 - z*x^k))^k, then a(n) ~ c * z^n, where c = Product_{j>=1} 1/(1 - 1/z^j)^(j+1).

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

Cf. A000219, A242587, A246935, A261561.

nmax = 40; CoefficientList[Series[Product[(1/(1 - 3*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[3^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

{a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, 3^d * m^2/d^2) ) +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 30 2015

a(n) ~ c * 3^n, where c = Product_{j>=1} 1/(1 - 1/3^j)^(j+1) = 4.1269357592430271005054028580646705856298720432004233223482475759761040273...

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} 3^d * n^2/d^2 ). - Paul D. Hanna, Sep 30 2015

A265974 Expansion of Product_{k>=1} 1/(1 - 3kx^k).

Original entry on oeis.org

1, 3, 15, 54, 210, 699, 2484, 7995, 26610, 84186, 269940, 839238, 2634579, 8098194, 25032282, 76388265, 233791104, 709501596, 2157488730, 6523204836, 19747491810, 59558682132, 179762506329, 541222906812, 1630300772106, 4902697929306, 14748249476553

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

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

Cf. A006906, A265951, A265975, A265976.

Cf. A246935, A242587.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      3^n, b(n, i-1) +i*3*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Aug 23 2019

nmax=40; CoefficientList[Series[Product[1/(1-3*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^n, where c = Product_{m>=2} 1/(1 - m/3^(m-1)) = 5.86277744540963226378877460838259757442241952947887939654316926419876...

A303390 Expansion of Product_{k>=1} (1 + 3x^k)/(1 - 3x^k).

Original entry on oeis.org

1, 6, 24, 96, 330, 1104, 3552, 11184, 34584, 105990, 322224, 975264, 2942016, 8857680, 26631312, 80005632, 240219114, 721036320, 2163789816, 6492625152, 19480105392, 58444390176, 175340344416, 526034008752, 1578124753152, 4734415061142, 14203316252400

Offset: 0

Vaclav Kotesovec, Apr 23 2018

nonn

Cf. A032308, A242587.

Cf. A015128, A261584, A303391.

nmax = 30; CoefficientList[Series[Product[(1+3*x^k)/(1-3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^n, where c = QPochhammer[-1, 1/3] / QPochhammer[1/3] = 5.5877920355220979147599292926505407983327527...

A352762 Expansion of Product_{k>=1} 1 / (1 + 3^(k-1)*x^k).

Original entry on oeis.org

1, -1, -2, -7, -11, -43, -65, -259, -146, -1798, 826, -8116, 17593, -35089, 301903, -308464, 3582403, 157367, 28816009, 9388694, 329375419, -61352008, 2991009094, 509592773, 23675224255, 1207374806, 229200996508, -129896994130, 2090952547882, -816324790165, 14079091274800

Offset: 0

Ilya Gutkovskiy, Jun 08 2022

sign

Cf. A032308, A242587, A261582, A292128, A300579, A344062, A352402, A352786.

nmax = 30; CoefficientList[Series[Product[1/(1 + 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[IntegerPartitions[n, {k}]] 3^(n - k), {k, 0, n}], {n, 0, 30}]

a(n) = Sum_{k=0..n} (-1)^k * p(n,k) * 3^(n-k), where p(n,k) is the number of partitions of n into k parts.

A370710 a(n) = 3^n * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/3).

Original entry on oeis.org

1, 3, 27, 180, 1431, 10206, 83025, 641277, 5264109, 42896790, 357649587, 2989185039, 25284805857, 214547921451, 1832454271926, 15702526829196, 135091225972926, 1165383100947105, 10081310266960155, 87401262194470719, 759320707197024909, 6608561546767471227, 57610976508944343963

Offset: 0

Vaclav Kotesovec, Feb 27 2024

nonn

Cf. A242587, A370712.

nmax = 25; CoefficientList[Series[Product[1/(1-3*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]
nmax = 25; CoefficientList[Series[Product[1/(1-3*(3*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[(-2/QPochhammer[3,x])^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]

G.f.: Product_{k>=1} 1/(1 - 3*(3*x)^k)^(1/3).

a(n) ~ c * 9^n / n^(2/3), where c = 1 / (Gamma(1/3) * QPochhammer(1/3)^(1/3)) = 0.45283708537555770181385241925945547307046394744...

A370714 a(n) = 4^n * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/2).

Original entry on oeis.org

1, 6, 78, 780, 8790, 90708, 1015692, 10964760, 122893926, 1370476932, 15518261220, 176063641512, 2014426860540, 23109736996680, 266397931733208, 3079014279154224, 35695144493030022, 414708043501061988, 4828444403991450612, 56314242827277224712, 657855733949279381652

Offset: 0

Vaclav Kotesovec, Feb 27 2024

nonn

Cf. A242587, A370711.

nmax = 25; CoefficientList[Series[Product[1/(1-3*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 25; CoefficientList[Series[Product[1/(1-3*(4*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[Sqrt[-2/QPochhammer[3,x]], {x, 0, nmax}], x] * 4^Range[0, nmax]

G.f.: Product_{k>=1} 1/(1 - 3*(4*x)^k)^(1/2).

a(n) ~ 12^n / sqrt(Pi*QPochhammer(1/3)*n).

A370735 a(n) = 5^(2n) [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/5).

Original entry on oeis.org

1, 15, 1050, 52125, 3277500, 179801250, 11966690625, 738318187500, 49788716718750, 3314446448437500, 227432073022265625, 15631633385109375000, 1090877899335878906250, 76338563689129101562500, 5384934139819611328125000, 381204340327212964599609375, 27111589537137988341064453125

Offset: 0

Vaclav Kotesovec, Feb 28 2024

nonn

In general, if d > 1, m >= 1 and g.f. = Product_{k>=1} 1/(1 - d*x^k)^(1/m), then a(n) ~ d^n / (Gamma(1/m) * QPochhammer(1/d)^(1/m) * n^(1 - 1/m)).

Cf. A242587 (d=3,m=1), A370714 (d=3,m=2), A370710 (d=3,m=3), A370734 (d=3,m=4).
Cf. A070933 (d=2,m=1), A370713 (d=2,m=2), A370715 (d=2,m=3), A370732 (d=2,m=4), A370733 (d=2,m=5).
Cf. A000041 (d=1,m=1), A271235 (d=1,m=2), A271236 (d=1,m=3).

nmax = 20; CoefficientList[Series[Product[1/(1-3*x^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x] * 25^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1/(1-3*(25*x)^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - 3*(25*x)^k)^(1/5).

a(n) ~ 75^n / (Gamma(1/5) * QPochhammer(1/3)^(1/5) * n^(4/5)).

cp's OEIS Frontend

A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A261582 Expansion of Product_{k>=1} 1/(1 + 3*x^k).

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A292128 Expansion of Product_{k>=1} (1 - 3*x^k).

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A261565 Expansion of Product_{k>=1} (1/(1 - 3*x^k))^k.

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A265974 Expansion of Product_{k>=1} 1/(1 - 3*k*x^k).

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A303390 Expansion of Product_{k>=1} (1 + 3*x^k)/(1 - 3*x^k).

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A352762 Expansion of Product_{k>=1} 1 / (1 + 3^(k-1)*x^k).

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A370710 a(n) = 3^n * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/3).

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A370714 a(n) = 4^n * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/2).

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A265974 Expansion of Product_{k>=1} 1/(1 - 3kx^k).

A303390 Expansion of Product_{k>=1} (1 + 3x^k)/(1 - 3x^k).

A370735 a(n) = 5^(2n) [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/5).