A003576 -id:A003576 - cp's OEIS frontend

A334162 a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (nk + 1)^n / (n^k k!).

1, 2, 6, 35, 352, 5307, 111592, 3117900, 111259904, 4912490375, 261954304224, 16560019685937, 1222893826048000, 104189533522270666, 10132262911996769408, 1114216450970154278543, 137427598621356912082944, 18877351974681584403701519, 2869969478954093766868948480

Offset: 0

Ilya Gutkovskiy, Apr 16 2020

nonn

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

Cf. A000110, A007405, A003575, A003576, A003577, A003578, A003579, A003580, A003581, A003582, A301419, A334165.

Table[SeriesCoefficient[1/(1 - x) Sum[(x/(1 - x))^k/Product[(1 - n j x/(1 - x)), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[n! SeriesCoefficient[Exp[x + (Exp[n x] - 1)/n], {x, 0, n}], {n, 1, 18}]]

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=0} (x/(1 - x))^k / Product_{j=1..k} (1 - n*j*x/(1 - x)).
a(n) = n! * [x^n] exp(x + (exp(n*x) - 1) / n), for n > 0.
a(n) = A334165(n,n).

A355167 a(n) = exp(-1/4) * Sum_{k>=0} (4k + 3)^n / (4^k k!).

Original entry on oeis.org

1, 4, 20, 128, 1008, 9280, 96704, 1120768, 14274816, 197833728, 2958521344, 47415508992, 809838505984, 14670950907904, 280760761434112, 5655835404271616, 119561580162646016, 2645030742360588288, 61087848487323959296, 1469652941137655103488, 36758243982057508175872, 954111239026567129595904

Offset: 0

Ilya Gutkovskiy, Jun 22 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..464
Adam Buck, Jennifer Elder, Azia A. Figueroa, Pamela E. Harris, Kimberly Harry, and Anthony Simpson, Flattened Stirling Permutations, arXiv:2306.13034 [math.CO], 2023. See p. 14.

Cf. A003576, A004213, A285064, A355164, A355165.

nmax = 21; CoefficientList[Series[Exp[3 x + (Exp[4 x] - 1)/4], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
Table[Sum[Binomial[n, k] 3^(n - k) 4^k BellB[k, 1/4], {k, 0, n}], {n, 0, 21}]

E.g.f.: exp(3*x + (exp(4*x) - 1) / 4).
a(0) = 1; a(n) = 3 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^(k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * A004213(k).
a(n) ~ 2^(2*n + 3/2) * n^(n + 3/4) * exp(n/LambertW(4*n) - n - 1/4) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^(n + 3/4)). - Vaclav Kotesovec, Jun 27 2022

A364069 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b = 63.

Original entry on oeis.org

1, 2, 67, 4355, 295234, 21036803, 1625419909, 140823067772, 13947448935109, 1570142163116087, 196457384808738412, 26717651072732512841, 3896182904620308595021, 605803757139146097600266, 100236348400243756326661039, 17619174544126256877550593743, 3280792242500933388439611444802

Offset: 0

Stefano Spezia, Jul 04 2023

nonn

a(n) is the number of all 64-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), A364070 (b=624).
Row sums of the triangle A364072.
2nd row of the array A364074.

With[{m=16, b=63}, 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]63^(n-d-k),{d,0,n-k}],{k,0,n}]; Array[a,17,0]

E.g.f.: exp(x + (exp(63*x) - 1)/63).
a(n) = exp(-1/63) * Sum_{k>=0} (63*k + 1)^n / (63^k * k!).
a(n) ~ 63^(n + 1/63) * n^(n + 1/63) * exp(n/LambertW(63*n) - n - 1/63) / (sqrt(1 + LambertW(63*n)) * LambertW(63*n)^(n + 1/63)).
a(n) = Sum_{k=0..n} Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*63^(n-d-k).

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

Original entry on oeis.org

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).

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).

A355165 a(n) = exp(-1/4) * Sum_{k>=0} (4k + 2)^n / (4^k k!).

Original entry on oeis.org

1, 3, 13, 79, 601, 5339, 53861, 607527, 7560625, 102637235, 1506225085, 23726435583, 398852249097, 7120170905995, 134408217821205, 2673140092099543, 55832167947587425, 1221199519275467107, 27902127744298836845, 664446811342185649583, 16457968670922936733113, 423242969435491221774907

Offset: 0

Ilya Gutkovskiy, Jun 22 2022

nonn

Cf. A003576, A004213, A285064, A355164, A355167.

nmax = 21; CoefficientList[Series[Exp[2 x + (Exp[4 x] - 1)/4], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
Table[Sum[Binomial[n, k] 2^(n + k) BellB[k, 1/4], {k, 0, n}], {n, 0, 21}]

E.g.f.: exp(2*x + (exp(4*x) - 1) / 4).
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^(k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * A004213(k).
a(n) ~ 2^(2*n+1) * n^(n + 1/2) * exp(n/LambertW(4*n) - n - 1/4) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^(n + 1/2)). - Vaclav Kotesovec, Jun 27 2022

cp's OEIS Frontend

A334162 a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (n*k + 1)^n / (n^k * k!).

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A355167 a(n) = exp(-1/4) * Sum_{k>=0} (4*k + 3)^n / (4^k * k!).

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A364069 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b = 63.

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

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Mathematica

Formula

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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Crossrefs

Programs

Mathematica

Formula

A355165 a(n) = exp(-1/4) * Sum_{k>=0} (4*k + 2)^n / (4^k * k!).

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Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A334162 a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (nk + 1)^n / (n^k k!).

A355167 a(n) = exp(-1/4) * Sum_{k>=0} (4k + 3)^n / (4^k k!).

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!).

A355165 a(n) = exp(-1/4) * Sum_{k>=0} (4k + 2)^n / (4^k k!).