A364072 -id:A364072 - cp's OEIS frontend

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

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).

A364073 Triangle read by rows: T(n, k) = Sum_{d=0..n-k} binomial(n, d)StirlingS2(n-d, k)624^(n-d-k), with 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 626, 1, 1, 391251, 1875, 1, 1, 244531876, 2733126, 3748, 1, 1, 152832422501, 3658206250, 9753130, 6245, 1, 1, 95520264063126, 4721932028751, 21925818740, 25346895, 9366, 1, 1, 59700165039453751, 5993213367973125, 45788990528771, 85217015555, 54578181, 13111, 1

Offset: 0

Stefano Spezia, Jul 04 2023

T(n, k) is the number of 625-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,          626,          1;
  1,       391251,       1875,       1;
  1,    244531876,    2733126,    3748,    1;
  1, 152832422501, 3658206250, 9753130, 6245, 1;
  ...

Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023. See pp. 7-9.

Cf. A000012 (k=0), A364070 (row sums), A364071, A364072.

T[n_,k_]:=Sum[Binomial[n,d]StirlingS2[n-d,k]624^(n-d-k),{d,0,n-k}]; Table[T[n,k],{n,0,7},{k,0,n}]//Flatten

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

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

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A364073 Triangle read by rows: T(n, k) = Sum_{d=0..n-k} binomial(n, d)StirlingS2(n-d, k)624^(n-d-k), with 0 <= k <= n.

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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.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A364073 Triangle read by rows: T(n, k) = Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*624^(n-d-k), with 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

A364073 Triangle read by rows: T(n, k) = Sum_{d=0..n-k} binomial(n, d)StirlingS2(n-d, k)624^(n-d-k), with 0 <= k <= n.

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.