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.

Previous Showing 21-25 of 25 results.

A337010 a(n) = exp(-1/2) * Sum_{k>=0} (2*k + 3)^n / (2^k * k!).

Original entry on oeis.org

1, 4, 18, 92, 532, 3440, 24552, 191280, 1612304, 14597952, 141123872, 1449324992, 15743376704, 180203389696, 2166381979264, 27274611880704, 358690234163456, 4916123783848960, 70076765972288000, 1036967662211324928, 15902394743591408640
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2020

Keywords

Crossrefs

Cf. A004211, A005494, A007405, A337011.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[3 x + (Exp[2 x] - 1)/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 4 a[n - 1] + Sum[Binomial[n - 1, k - 1] 2^(k - 1) a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 20}]

Formula

E.g.f.: exp(3*x + (exp(2*x) - 1) / 2).
a(0) = 1; a(n) = 4 * a(n-1) + Sum_{k=2..n} binomial(n-1,k-1) * 2^(k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * A004211(k+1).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * A004211(k).

A111670 Array T(n,k) read by antidiagonals: the k-th column contains the first column of the k-th power of A039755.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 15, 24, 1, 1, 5, 28, 105, 116, 1, 1, 6, 45, 280, 929, 648, 1, 1, 7, 66, 585, 3600, 9851, 4088, 1, 1, 8, 91, 1056, 9865, 56240, 121071, 28640, 1
Offset: 1

Views

Author

Gary W. Adamson, Aug 14 2005

Keywords

Examples

			 1     1       1          1          1          1          1          1
 1     2       3          4          5          6          7          8
 1     6      15         28         45         66         91        120
 1    24     105        280        585       1056       1729       2640
 1   116     929       3600       9865      22036      43001      76224
 1   648    9851      56240     203565     565096    1318023    2717856
 1  4088  121071    1029920    4953205   17148936   47920803  115146816
 1 28640 1685585   21569600  138529105  600001696 2012844225 5644055040

Crossrefs

Cf. A039755, A007405 (column 2), A000384 (row 2), A011199 (row 3).

Programs

  • Maple
    A111670 := proc(n,k)
    local A,i,j ;
    A := Matrix(n,n) ;
    for i from 1 to n do
    for j from 1 to n do
        A[i,j] := A039755(i-1,j-1) ;
    end do:
    end do:
    LinearAlgebra[MatrixPower](A,k) ;
    %[n,1] ;
end proc:
for d from 2 to 12 do
    for n from  1 to d-1 do
        printf("%d,",A111670(n,d-n)) ;
    end do:
end do: # R. J. Mathar, Jan 27 2023
  • Mathematica
    nmax = 10;
A[n_, k_] := Sum[(-1)^(k-j)*(2j+1)^n*Binomial[k, j], {j, 0, k}]/(2^k*k!);
A039755 = Array[A, {nmax, nmax}, {0, 0}];
T = Table[MatrixPower[A039755, n][[All, 1]], {n, 1, nmax}] // Transpose;
Table[T[[n-k+1, k]], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Apr 02 2024 *)

Formula

Let A039755 (an analog of Stirling numbers of the second kind) be an infinite lower triangular matrix M; then the vector M^k * [1, 0, 0, 0, ...] (first column of the k-th power) is the k-th column of this array.

Extensions

Definition simplified by R. J. Mathar, Jan 27 2023

A363908 a(n) = exp(-1/5) * Sum_{k>=0} (5*k + 4)^n / (5^k * k!).

Original entry on oeis.org

1, 5, 30, 225, 2075, 22500, 276875, 3790625, 57050000, 934984375, 16549046875, 314146406250, 6358972578125, 136603266015625, 3101556258593750, 74164388642578125, 1861859526474609375, 48936176077929687500, 1343302192888037109375, 38425435693841064453125, 1143143051078878906250000
Offset: 0

Views

Author

Michael De Vlieger, Jun 27 2023

Keywords

Links

Crossrefs

Cf. A007405, A050488, A355164, A355167.

Programs

  • Mathematica
    CoefficientList[Series[Exp[4 x + (Exp[5 x] - 1)/5], {x, 0, #}], x]* Range[0, #]! &[21]

A039764 D-analogs of Bell numbers.

Original entry on oeis.org

1, 1, 4, 15, 72, 403, 2546, 17867, 137528, 1149079, 10335766, 99425087, 1017259964, 11018905667, 125860969266, 1510764243699, 18999827156304, 249687992188015, 3420706820299374, 48751337014396167
Offset: 0

Views

Author

Ruedi Suter (suter(AT)math.ethz.ch)

Keywords

Links

Crossrefs

B-analogs of Bell numbers = A007405.

Programs

  • Mathematica
    Range[0, 25]! CoefficientList[Series[(Exp[x] - x) Exp[1/2 (Exp[2 x] - 1)], {x, 0, 25}], x] (* Vincenzo Librandi, May 03 2015 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace((exp(x) - x)*exp(1/2*(exp(2*x) - 1)))) \\ Michel Marcus, May 03 2015

Formula

E.g.f.: (exp(x) - x)*exp(1/2*(exp(2*x) - 1)).
a(n) = Sum_{k=0..n} A039760(n, k).

A337012 a(n) = exp(-1/2) * Sum_{k>=0} (2*k + n)^n / (2^k * k!).

Original entry on oeis.org

1, 2, 11, 92, 1025, 14232, 236403, 4568720, 100670529, 2490511776, 68341981051, 2059882505408, 67645498798721, 2403948686290816, 91914992104815459, 3762299973887526144, 164148252324092964993, 7604537914425558921728, 372812121514187124192875
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2020

Keywords

Crossrefs

Cf. A004211, A007405, A134980, A337010, A337011.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[n x + (Exp[2 x] - 1)/2], {x, 0, n}], {n, 0, 18}]
Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k] n^(n - k) 2^k BellB[k, 1/2], {k, 0, n}], {n, 0, 18}]

Formula

a(n) = n! * [x^n] exp(n*x + (exp(2*x) - 1) / 2).
a(n) = Sum_{k=0..n} binomial(n,k) * n^(n-k) * A004211(k).
Previous Showing 21-25 of 25 results.

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