A070933 -id:A070933 - cp's OEIS frontend

A346848 Number of conjugacy classes of the symplectic group Sp(2n, 2) over the field with 2 elements.

1, 3, 11, 30, 81, 198, 477, 1089, 2451, 5358, 11567, 24537, 51577, 107205, 221378, 453900, 926395, 1882152, 3812232, 7699191, 15518112, 31220991, 62733296, 125911851, 252516626, 506082933, 1013780968, 2029989807, 4063678159, 8132877129, 16274093175

Offset: 0

Jan Kristian Haugland, Aug 06 2021

nonn

Sp(2n, 2) is isomorphic to the orthogonal group O(2n+1, 2) over the field with 2 elements, and is a simple and complete group for n>=3.

			a(2)=11, and Sp(4, 2) is isomorphic to the symmetric group S_6 which has 11 conjugacy classes.

Jan Kristian Haugland, Table of n, a(n) for n = 0..1000
G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Aust. Math. Soc., 3 (1963), 1-62.

Discrete convolution of A070933 and A098613. A003923 gives the order of the group.

A352402 Expansion of Product_{k>=1} 1 / (1 + 2^(k-1)*x^k).

Original entry on oeis.org

1, -1, -1, -3, -1, -7, -1, -15, 31, -63, 159, -95, 671, -287, 3231, -2975, 15519, -7839, 44191, -34975, 224415, -291999, 863391, -990367, 2927775, -4902047, 12561567, -27225247, 56470687, -102640799, 152153247, -422620319, 877243551, -2278272159, 3357125791

Offset: 0

Ilya Gutkovskiy, Jun 08 2022

sign

Cf. A032302, A070877, A070933, A071109, A075900, A304961, A327550, A337299, A352762.

nmax = 34; CoefficientList[Series[Product[1/(1 + 2^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[IntegerPartitions[n, {k}]] 2^(n - k), {k, 0, n}], {n, 0, 34}]

a(n) = Sum_{k=0..n} (-1)^k * p(n,k) * 2^(n-k), where p(n,k) is the number of partitions of n into k parts.

A355390 Number of ordered pairs of distinct integer partitions of n.

Original entry on oeis.org

0, 0, 2, 6, 20, 42, 110, 210, 462, 870, 1722, 3080, 5852, 10100, 18090, 30800, 53130, 87912, 147840, 239610, 392502, 626472, 1003002, 1573770, 2479050, 3831806, 5931660, 9057090, 13819806, 20834660, 31399212, 46806122, 69697452, 102870306, 151523790, 221488806

Offset: 0

Gus Wiseman, Jul 04 2022

nonn

			The a(0) = 0 through a(3) = 6 pairs:
  .  .  (11)(2)  (21)(3)
        (2)(11)  (3)(21)
                 (111)(3)
                 (3)(111)
                 (111)(21)
                 (21)(111)

Without distinctness we have A001255, unordered A086737.
The version for compositions is A020522, unordered A006516.
The unordered version is A355389.
A000041 counts partitions, strict A000009.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Cf. A022811, A070933, A316245, A317715, A319910, A323433, A323583, A339006, A355385.

Table[Length[Select[Tuples[IntegerPartitions[n],2],UnsameQ@@#&]],{n,0,15}]

a(n) = 2*binomial(numbpart(n), 2); \\ Michel Marcus, Jul 05 2022

a(n) = 2*A355389(n) = 2*binomial(A000041(n), 2).

A371546 Expansion of e.g.f. Product_{k>=1} 1 / (1 - 2*x^k/k!).

Original entry on oeis.org

1, 2, 10, 62, 522, 5262, 64006, 897990, 14416618, 259650638, 5197438710, 114360488310, 2745242514966, 71378953200310, 1998718342001062, 59962112293963182, 1918813454880552298, 65239810516299767310, 2348641102002493520086, 89248414267689180772278, 3569939582019832830181222

Offset: 0

Ilya Gutkovskiy, Mar 27 2024

nonn

Cf. A005651, A032297, A070933.

nmax = 20; CoefficientList[Series[Product[1/(1 - 2 x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

a(n) ~ c * 2^n * n!, where c = Product_{k>=2} 1/(1 - 2^(1-k)/k!) = 1.39938283723373672673056837661175942499559257652969647531100283042201554... - Vaclav Kotesovec, Mar 28 2024

A329274 Expansion of 1 / (1 + Sum_{k>=1} phi(k) * log(1 - 2 * x^k) / k), where phi = A000010.

Original entry on oeis.org

1, 2, 7, 24, 83, 286, 989, 3416, 11807, 40806, 141041, 487488, 1684971, 5823986, 20130299, 69579356, 240497727, 831269134, 2873243541, 9931234972, 34326861907, 118649239730, 410105717339, 1417511828340, 4899565424887, 16935125993974, 58535496103303, 202325291692972

Offset: 0

Ilya Gutkovskiy, Nov 11 2019

nonn

Invert transform of A000031.

Cf. A000010, A000031, A070933.

nmax = 27; CoefficientList[Series[1/(1 + Sum[EulerPhi[k] Log[1 - 2 x^k]/k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(1/k) DivisorSum[k, EulerPhi[#] 2^(k/#) &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 27}]

a(0) = 1; a(n) = Sum_{k=1..n} A000031(k) * a(n-k).

cp's OEIS Frontend

A346848 Number of conjugacy classes of the symplectic group Sp(2n, 2) over the field with 2 elements.

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A352402 Expansion of Product_{k>=1} 1 / (1 + 2^(k-1)*x^k).

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A355390 Number of ordered pairs of distinct integer partitions of n.

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A371546 Expansion of e.g.f. Product_{k>=1} 1 / (1 - 2*x^k/k!).

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A329274 Expansion of 1 / (1 + Sum_{k>=1} phi(k) * log(1 - 2 * x^k) / k), where phi = A000010.

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