A003580 -id:A003580 - cp's OEIS frontend

A364070 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=624.

1, 2, 628, 393128, 247268752, 156500388128, 100264147266880, 65739252669562496, 44949841635462426880, 32961816599696140935680, 26763226019573589904012288, 24577197816669853786615064576, 25455086256328481246829666144256, 29063231104986184254344094194278400

Offset: 0

Stefano Spezia, Jul 04 2023

nonn

a(n) is the number of all 625-subgroups of R^n, where R^n is a near-vector space over a proper nearfield R.

Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023. See pp. 7-8.
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.

Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10), A364069 (b=63).
Row sums of the triangle A364073.
3rd row of the array A364074.

With[{m=13, b=624}, CoefficientList[Series[Exp[x +(Exp[b*x]-1)/b], {x, 0, m}], x]*Range[0, m]!] (* or *)
a[n_]:=Sum[Sum[Binomial[n,d]StirlingS2[n-d,k]624^(n-d-k),{d,0,n-k}],{k,0,n}]; Array[a,14,0]

E.g.f.: exp(x + (exp(624*x) - 1)/624).
a(n) = exp(-1/624) * Sum_{k>=0} (624*k + 1)^n / (624^k * k!).
a(n) ~ 624^(n + 1/624) * n^(n + 1/624) * exp(n/LambertW(624*n) - n - 1/624) / (sqrt(1 + LambertW(624*n)) * LambertW(624*n)^(n + 1/624)).
a(n) = Sum_{k=0..n} Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*624^(n-d-k).

A364074 Array read by ascending antidiagonals: A(m, n) = Sum_{i=0..n} Sum_{d=0..n-i} binomial(n, d)StirlingS2(n-d, i)(m^(m-1) - 1)^(n-d-i).

Original entry on oeis.org

1, 1, 2, 1, 2, 12, 1, 2, 67, 120, 1, 2, 628, 4355, 1424, 1, 2, 7779, 393128, 295234, 19488, 1, 2, 117652, 60497283, 247268752, 21036803, 307904, 1, 2, 2097155, 13841757800, 470668752866, 156500388128, 1625419909, 5539712, 1, 2, 43046724, 4398054899715, 1628524328796304, 3663682367243907, 100264147266880, 140823067772, 111259904

Offset: 2

Stefano Spezia, Jul 04 2023

A(m, n) is the number of all ((m+1)^m)-subgroups of R^n, where R^n is a near-vector space over a proper nearfield R.

			The array begins:
  1, 2,   12,      120,         1424,            19488, ...
  1, 2,   67,     4355,       295234,         21036803, ...
  1, 2,  628,   393128,    247268752,     156500388128, ...
  1, 2, 7779, 60497283, 470668752866, 3663682367243907, ...
  ...

Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023. See pp. 7-8.

Cf. A003580 (m=2), A364069 (m=3), A364070 (m=4), A364075 (antidiagonal sums).

A[m_,n_]:=Sum[Sum[Binomial[n,d]StirlingS2[n-d,i](m^(m-1)-1)^(n-d-i),{d,0,n-i}],{i,0,n}]; Table[A[m-n+1,n],{m,2,10},{n,0,m-2}]//Flatten

A334165 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(-1/k) * Sum_{j>=0} (kj + 1)^n / (k^j j!).

Original entry on oeis.org

1, 1, 2, 1, 2, 5, 1, 2, 6, 15, 1, 2, 7, 24, 52, 1, 2, 8, 35, 116, 203, 1, 2, 9, 48, 214, 648, 877, 1, 2, 10, 63, 352, 1523, 4088, 4140, 1, 2, 11, 80, 536, 3008, 12349, 28640, 21147, 1, 2, 12, 99, 772, 5307, 29440, 112052, 219920, 115975, 1, 2, 13, 120, 1066, 8648, 60389, 324096, 1120849, 1832224, 678570

Offset: 0

Ilya Gutkovskiy, Apr 17 2020

Square array of Dowling numbers.

			Square array begins:
    1,    1,     1,     1,     1,     1,  ...
    2,    2,     2,     2,     2,     2,  ...
    5,    6,     7,     8,     9,    10,  ...
   15,   24,    35,    48,    63,    80,  ...
   52,  116,   214,   352,   536,   772,  ...
  203,  648,  1523,  3008,  5307,  8648,  ...

Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.

Columns k=1..10 give A000110 (for n > 0), A007405, A003575, A003576, A003577, A003578, A003579, A003580, A003581, A003582.

Cf. A241578, A241579, A334162 (diagonal).

Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[(x/(1 - x))^j/Product[(1 - k i x/(1 - x)), {i, 1, j}], {j, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten
Table[Function[k, n! SeriesCoefficient[Exp[x + (Exp[k x] - 1)/k], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten

G.f. of column k: (1/(1 - x)) * Sum_{j>=0} (x/(1 - x))^j / Product_{i=1..j} (1 - k*i*x/(1 - x)).

E.g.f. of column k: exp(x + (exp(k*x) - 1) / k).

A364071 Triangle read by rows: T(n, k) = Sum_{d=0..n-k} binomial(n, d)StirlingS2(n-d, k)8^(n-d-k), with 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 91, 27, 1, 1, 820, 550, 52, 1, 1, 7381, 10170, 1850, 85, 1, 1, 66430, 180271, 56420, 4655, 126, 1, 1, 597871, 3131037, 1590771, 210035, 9821, 175, 1, 1, 5380840, 53825500, 42900312, 8521926, 612696, 18396, 232, 1, 1, 48427561, 920414340, 1126333300, 324123870, 33642462, 1514100, 31620, 297, 1

Offset: 0

Stefano Spezia, Jul 04 2023

T(n, k) is the number of 8-subgroups of R^n which have dimension k, where R^n is a near-vector space over a proper nearfield R.

			The triangle begins:
  1;
  1,      1;
  1,     10,       1;
  1,     91,      27,      1;
  1,    820,     550,     52,       1;
  1,   7381,   10170,   1850,      85,    1;
  1,  66430,  180271,  56420,    4655,  126,   1;
  1, 597871, 3131037, 1590771, 210035, 9821, 175, 1;
  ...

Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023. See pp. 7-8.

Cf. A000012 (k=0), A002452 (k=1), A003580 (row sums), A364072, A364073.

T[n_,k_]:=Sum[Binomial[n,d]StirlingS2[n-d,k]8^(n-d-k),{d,0,n-k}]; Table[T[n,k],{n,0,9},{k,0,n}]//Flatten

cp's OEIS Frontend

A364070 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=624.

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A364074 Array read by ascending antidiagonals: A(m, n) = Sum_{i=0..n} Sum_{d=0..n-i} binomial(n, d)*StirlingS2(n-d, i)*(m^(m-1) - 1)^(n-d-i).

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A334165 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(-1/k) * Sum_{j>=0} (k*j + 1)^n / (k^j * j!).

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Author

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Comments

Examples

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Programs

Mathematica

Formula

A364071 Triangle read by rows: T(n, k) = Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*8^(n-d-k), with 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A364074 Array read by ascending antidiagonals: A(m, n) = Sum_{i=0..n} Sum_{d=0..n-i} binomial(n, d)StirlingS2(n-d, i)(m^(m-1) - 1)^(n-d-i).

A334165 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(-1/k) * Sum_{j>=0} (kj + 1)^n / (k^j j!).

A364071 Triangle read by rows: T(n, k) = Sum_{d=0..n-k} binomial(n, d)StirlingS2(n-d, k)8^(n-d-k), with 0 <= k <= n.