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.

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A317171 a(n) = n! * [x^n] 1/(1 + n*log(1 - x)).

Original entry on oeis.org

1, 1, 10, 222, 8824, 553870, 50545008, 6328330344, 1041597412224, 218138133235680, 56650689388344000, 17868469522986145536, 6728682216722958185472, 2981868816113406609186576, 1536217706761623823662025728, 910442461680276910819097616000, 615053979239579281793375485526016
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 23 2018

Keywords

Crossrefs

Cf. A007840, A088500, A094420, A242817, A317172.
Main diagonal of A320079.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(1 + n Log[1 - x]), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[Abs[StirlingS1[n, k]] n^k k!, {k, n}], {n, 16}]]

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*n^k*k!.
a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n - 1/2). - Vaclav Kotesovec, Jul 23 2018

A335867 a(n) = exp(-n) * Sum_{k>=0} n^k * (k - 1)^n / k!.

Original entry on oeis.org

1, 0, 3, 29, 397, 6879, 144751, 3587100, 102351929, 3305310065, 119186370091, 4746969337923, 206966647324933, 9804683604806908, 501491905963040903, 27544070654283355889, 1616869985889305862385, 101020181695996141703335, 6693303018177050431484035, 468770856837303230888704208
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2020

Keywords

Links

Crossrefs

Cf. A000296, A217924, A242817, A334240, A335868.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[n (Exp[x] - 1) - x], {x, 0, n}], {n, 0, 19}]
Table[Sum[(-1)^(n - k) Binomial[n, k] BellB[k, n], {k, 0, n}], {n, 0, 19}]

Formula

a(n) = n! * [x^n] exp(n*(exp(x) - 1) - x).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * BellPolynomial_k(n).

A346654 a(n) = Bell(2*n,n).

Original entry on oeis.org

1, 2, 94, 12351, 3188340, 1362057155, 869725707522, 775929767223352, 921839901090823112, 1406921223577401454239, 2682502220690005671884710, 6248503930824315386034050253, 17460431497766377837983159782652, 57647207262184459310081410522242310, 222006095854149044448961838142906736554
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 27 2021

Keywords

Links

Crossrefs

Cf. A000110, A189233, A242817, A292860, A346655.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, j-1)*b(n-j, k), j=1..n-1))*k)
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..14);  # Alois P. Heinz, Jul 27 2021
  • Mathematica
    Table[BellB[2*n, n], {n, 0, 20}]

Formula

a(n) ~ 4^n * exp((2/LambertW(2) - 3)*n) * n^(2*n) / (sqrt(1 + LambertW(2)) * LambertW(2)^(2*n)).
a(n) = A189233(2n,n) = A292860(2n,n). - Alois P. Heinz, Jul 27 2021

A346655 a(n) = Bell(3*n,n).

Original entry on oeis.org

1, 5, 2430, 5597643, 35618229364, 483040313859705, 11977437107679230274, 490630568583958198181583, 30889771581097736768046865352, 2832037863467651034046820871428061, 362579939205426756198837321528946171110, 62687814132880422794200073791149602981717667
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 27 2021

Keywords

Comments

In general, for k>=1, Bell(k*n,n) ~ (k*n/LambertW(k))^(k*n) / (sqrt(1 + LambertW(k)) * exp(n*(k + 1 - k/LambertW(k)))).

Links

Crossrefs

Cf. A000110, A189233, A242817, A292860, A346654.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, j-1)*b(n-j, k), j=1..n-1))*k)
    end:
a:= n-> b(3*n, n):
seq(a(n), n=0..11);  # Alois P. Heinz, Jul 27 2021
  • Mathematica
    Table[BellB[3*n, n], {n, 0, 15}]

Formula

a(n) ~ (3*n/LambertW(3))^(3*n) / (sqrt(1 + LambertW(3)) * exp(n*(4 - 3/LambertW(3)))).
a(n) = A189233(3n,n) = A292860(3n,n). - Alois P. Heinz, Jul 27 2021

A357682 a(n) = Sum_{k=0..floor(n/2)} n^k * Stirling2(n,2*k).

Original entry on oeis.org

1, 0, 2, 9, 44, 325, 2742, 24794, 250168, 2796795, 33842610, 439337085, 6100179780, 90139379928, 1409779442190, 23242554452745, 402652762232048, 7308371248274949, 138605556986785674, 2740167375732394378, 56350604098768558140, 1203156656491936711635
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2022

Keywords

Links

Crossrefs

Main diagonal of A357681.
Cf. A242817, A357683.

Programs

  • PARI
    a(n) = sum(k=0, n\2, n^k*stirling(n, 2*k, 2));
  • PARI
    a(n) = round(n!*polcoef(cosh(sqrt(n)*(exp(x+x*O(x^n))-1)), n));
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(n))+Bell_poly(n, -sqrt(n))))/2;

Formula

a(n) = n! * [x^n] cosh( sqrt(n) * (exp(x) - 1) ).
a(n) = ( Bell_n(sqrt(n)) + Bell_n(-sqrt(n)) )/2, where Bell_n(x) is n-th Bell polynomial.

A295623 a(n) = n! * [x^n] exp(n*x*exp(x)).

Original entry on oeis.org

1, 1, 8, 90, 1424, 28900, 716292, 20972098, 708317248, 27108056808, 1159375192100, 54799938951934, 2836735081572240, 159606310760007436, 9698172715195196260, 632924646574215596850, 44153807025286701187328, 3278903858941755472870864, 258247909552273997037934788
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 24 2017

Keywords

Links

Crossrefs

Cf. A000248, A242817, A275707, A351765.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[n x Exp[x]], {x, 0, n}], {n, 0, 18}]
Table[Sum[BellY[n, k, n Range[n]], {k, 0, n}], {n, 0, 18}]
  • PARI
    a(n) = sum(k=0, n, n^k*k^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Jul 04 2022

Formula

a(n) = n! * [x^n] exp(n*Sum_{k>=1} x^k/(k - 1)!).
From Seiichi Manyama, Jul 05 2022: (Start)
a(n) = [x^n] Sum_{k>=0} (n * x)^k/(1 - k*x)^(k+1).
a(n) = Sum_{k=0..n} n^k * k^(n-k) * binomial(n,k). (End)

A338282 a(n) = (1/e^n) * Sum_{j>=3} j^n * n^j / (j-3)!.

Original entry on oeis.org

0, 4, 216, 7371, 239424, 8127875, 296315496, 11685617608, 498593804800, 22959117809685, 1137033860419000, 60338078785131785, 3418430599382500800, 206053517402599981504, 13172124530670958537160, 890361160360138336174875, 63463906792476058870550528, 4758276450884470061869230823
Offset: 0

Views

Author

Pedro Caceres, Oct 20 2020

Keywords

Examples

			a(3) = 7371 = (1/e^3) * Sum_{j>=3} j^3 * 3^j / factorial(j-3).

Crossrefs

Cf. A005494, A143495, A242817, A299824.

Programs

  • Maple
    seq(add(n^(k+3)*A143495(n+3, k+3), k = 0..n), n = 0..17); # Peter Luschny, Oct 21 2020
  • Mathematica
    a[n_] := Exp[-n] * Sum[j^n * n^j/(j - 3)!, {j, 3, Infinity}]; Array[a, 17, 0] (* Amiram Eldar, Oct 20 2020 *)
  • PARI
    a(n)={sum(k=0, n+3, n^k*(stirling(n+3,k,2) - 3*stirling(n+2,k,2) + 2*stirling(n+1,k,2)))} \\ Andrew Howroyd, Oct 20 2020
  • SageMath
    # Increase precision for larger n!
R = RealField(100)
t = 3
sol = [0]*18
for n in range(0, 18):
    suma = R(0)
    for j in range(t, 1000):
        suma += (j^n * n^j) / factorial(j - t)
    suma *= exp(-n)
    sol[n] = round(suma)
print(sol) # Peter Luschny, Oct 20 2020

Formula

a(n) = Sum_{k=0..n+3} n^k*(Stirling2(n+3,k) - 3*Stirling2(n+2,k) + 2*Stirling2(n+1,k)). - Andrew Howroyd, Oct 20 2020
a(n) = Sum_{k=0..n} n^(k+3)*A143495(n+3, k+3). - Peter Luschny, Oct 21 2020

A339401 a(n) = numerator of (1/e)^n * Sum_{k>=0}(n^k*k^n)/(n!*k!).

Original entry on oeis.org

1, 1, 3, 19, 63, 322, 44683, 941977, 4677605, 668520163, 21622993111, 9759873853, 31135480907413, 194137920764803, 64440211018897379, 3298807094967155971, 181322497435007616497, 532556590750629416219, 665881649529214120845679, 2596711638295426703997397, 1031081559092352146579024047
Offset: 0

Views

Author

William C. Laursen, Dec 03 2020

Keywords

Crossrefs

Cf. A242817, A339402 (denominators).

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, (1+
      add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
a:= n-> numer(A(n$2)/n!):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 07 2020
  • Mathematica
    a[n_] := BellB[n, n]/n! // Numerator;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 27 2022 *)

Formula

a(n)/A339402(n) = A242817(n)/n!. - Pontus von Brömssen, Dec 03 2020
a(n) = numerator([x^n] exp(n*(exp(x)-1))). - Alois P. Heinz, Dec 07 2020

A339402 a(n) = denominator of (1/e)^n * Sum_{k>=0}(n^k*k^n)/(n!*k!).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 120, 720, 1008, 40320, 362880, 45360, 39916800, 68428800, 6227020800, 87178291200, 1307674368000, 1046139494400, 355687428096000, 376610217984000, 40548366802944000, 2432902008176640000, 5676771352412160000, 40142883134914560000, 25852016738884976640000
Offset: 0

Views

Author

William C. Laursen, Dec 03 2020

Keywords

Crossrefs

Cf. A339401 for numerators and relation to A242817.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, (1+
      add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
a:= n-> denom(A(n$2)/n!):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 07 2020
  • Mathematica
    a[n_] := BellB[n, n]/n! // Denominator;
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 27 2022 *)

Formula

A339401(n)/a(n) = A242817(n)/n!. - Pontus von Brömssen, Dec 03 2020
a(n) = denominator([x^n] exp(n*(exp(x)-1))). - Alois P. Heinz, Dec 07 2020

A332188 a(n) = (1/e^n) * Sum_{j>=2} j^n * n^j / (j-2)!.

Original entry on oeis.org

0, 3, 72, 1557, 36928, 986550, 29641608, 994006209, 36887753216, 1502798312547, 66730937637400, 3209318261685690, 166242143849148864, 9229638177763268395, 546842961612529341032, 34443269219453881669425
Offset: 0

Views

Author

Pedro Caceres, Oct 30 2020

Keywords

Examples

			a(3) = 1557 = (1/e^3) * Sum_{j>=2} j^3 * 3^j / factorial(j-2).

Crossrefs

Cf. A005494, A143495, A242817, A299824, A338282.

Programs

  • Mathematica
    a[n_] := Sum[n^k*(StirlingS2[n + 2, k] - StirlingS2[n + 1, k]), {k, 2, n + 2}]; Array[a, 16, 0] (* Amiram Eldar, Oct 30 2020 *)
  • PARI
    a(n) = sum(k=0, n+2, n^k*(stirling(n+2,k,2) - stirling(n+1,k,2))); \\ Michel Marcus, Oct 30 2020
  • SageMath
    # Increase precision for larger n!
R = RealField(100)
t = 2
sol = [0]*18
for n in range(0, 18):
    suma = R(0)
    for j in range(t, 1000):
        suma += (j^n * n^j) / factorial(j - t)
    suma *= exp(-n)
    sol[n] = round(suma)
print(sol) # Thanks to Peter Luschny for his example in A338282.

Formula

a(n) = Sum_{k=0..n+2} n^k*(Stirling2(n+2,k) - Stirling2(n+1,k)). [Thanks to Andrew Howroyd for his example in A338282]
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