cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A003581 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=9.

Original entry on oeis.org

1, 2, 13, 143, 1852, 27563, 473725, 9290396, 203745235, 4912490375, 128777672338, 3643086083981, 110557605978901, 3579776914324250, 123074955978249433, 4474133111905169219, 171363047274358839412, 6893620459732188296591, 290475101469031118494993
Offset: 0

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Author

N. J. A. Sloane

Keywords

Examples

			G.f. = 1 + 2*x + 13*x^2 + 143*x^3 + 1852*x^4 + 27563*x^5 + ...

Links

Crossrefs

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), this sequence (b=9), A003582 (b=10), A364069 (b=63), A364070 (b=624).

Programs

  • Magma
    m:=20; c:=9; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +(Exp(c*x)-1)/c) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 24 2019
  • Maple
    seq(coeff(series(factorial(n)*exp(z+(1/9)*exp(9*z)-(1/9)),z,n+1), z, n), n = 0 .. 20); # Muniru A Asiru, Feb 24 2019
  • Mathematica
    With[{m=20, b=9}, CoefficientList[Series[Exp[x +(Exp[b*x]-1)/b],{x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Feb 24 2019 *)
Table[Sum[Binomial[n, k] * 9^k * BellB[k, 1/9], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)
  • PARI
    Vec(serlaplace(exp(z + (exp(9*z) - 1)/9))) \\ Michel Marcus, Nov 07 2014
  • Sage
    m = 20; b=9; T = taylor(exp(x +(exp(b*x)-1)/b), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 24 2019

Formula

E.g.f.: exp(x + (exp(9*x) - 1)/9).
G.f.: 1/W(0), where W(k) = 1 - x - x/(1 - 9*(k+1)*x/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 07 2014
a(n) = exp(-1/9) * Sum_{k>=0} (9*k + 1)^n / (9^k * k!). - Ilya Gutkovskiy, Apr 16 2020
a(n) ~ 9^(n + 1/9) * n^(n + 1/9) * exp(n/LambertW(9*n) - n - 1/9) / (sqrt(1 + LambertW(9*n)) * LambertW(9*n)^(n + 1/9)). - Vaclav Kotesovec, Jun 26 2022

Extensions

Name clarified by Muniru A Asiru, Feb 24 2019

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

Views

Author

Stefano Spezia, Jul 04 2023

Keywords

Comments

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.

Links

Crossrefs

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.

Programs

  • Mathematica
    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]

Formula

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

Views

Author

Stefano Spezia, Jul 04 2023

Keywords

Comments

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.

Examples

			The triangle begins:
  1;
  1,            1;
  1,          626,          1;
  1,       391251,       1875,       1;
  1,    244531876,    2733126,    3748,    1;
  1, 152832422501, 3658206250, 9753130, 6245, 1;
  ...

Links

Crossrefs

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

Programs

  • Mathematica
    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

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

Views

Author

Stefano Spezia, Jul 04 2023

Keywords

Comments

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.

Examples

			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, ...
  ...

Links

Crossrefs

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

Programs

  • Mathematica
    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
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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