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 1-5 of 5 results.

A321837 Expansion of e.g.f.: exp(x/(1-3*x)).

Original entry on oeis.org

1, 1, 7, 73, 1009, 17341, 355951, 8488117, 230439553, 7013527129, 236419161751, 8740611892321, 351566026652017, 15280473017519893, 713558666964639679, 35623071889296787981, 1893073661362838712961, 106682309871314293118257
Offset: 0

Views

Author

Ludovic Schwob, Nov 19 2018

Keywords

Comments

For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.

Links

Crossrefs

Cf. A000262, A025168, A321847, A321848, A321849, A321850 (analogs for k=1,2,4,5,6,7).

Programs

  • GAP
    Concatenation([1], List([1..25], n-> Sum([1..n], k-> 3^(n-k)*(Factorial(n)/Factorial(k))*Binomial(n-1, k-1)))); # G. C. Greubel, Dec 14 2018
  • Magma
    [1] cat [&+[3^(n-k)*Factorial(n) div Factorial(k)*Binomial(n-1, k-1):  k in [0..n]]: n in [1.. 18]]; // Vincenzo Librandi, Dec 08 2018
  • Maple
    seq(coeff(series(factorial(n)*exp(x/(1-3*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018
  • Mathematica
    a[n_] := Sum[3^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (6n - 5)*a[n - 1] - 9(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)
  • PARI
    my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-3*x)))) \\ Michel Marcus, Nov 25 2018
  • Sage
    {c[1]:c[0]*factorial(c[1]) for c in (exp(x/(1-3*x))).taylor(x,0,25).coefficients()} # G. C. Greubel, Dec 14 2018

Formula

a(n) = Sum_{k=0..n} 3^(n-k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (6*n-5)*a(n-1) - 9*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(2*sqrt(n/3) - 1/6) * 3^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018

A321847 E.g.f.: exp(x/(1 - 4*x)).

Original entry on oeis.org

1, 1, 9, 121, 2161, 48081, 1279801, 39631369, 1398961761, 55422807841, 2434261023721, 117366299630361, 6161301265353169, 349768597919934961, 21347094823271661081, 1393695557886847095721, 96910923898115717350081, 7149718240571434690591809
Offset: 0

Views

Author

Ludovic Schwob, Nov 19 2018

Keywords

Comments

For k = 2, 3, 4, ... the difference a(n+k) - a(n) is divisible by k.

Links

Crossrefs

Cf. A000262, A025168, A321837, A321848, A321849, A321850.

Programs

  • Maple
    seq(coeff(series(factorial(n)*exp(x/(1-4*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018
  • Mathematica
    a[n_] := Sum[4^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (8n - 7)*a[n - 1] - 16(n - 2)(n - 1) *a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)
CoefficientList[Series[Exp[x/(1 - 4*x)], {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Dec 07 2018 *)
  • Maxima
    (a[0] : 1, a[1] : 1, a[n] := (8*n - 7)*a[n-1] - 16*(n-2)*(n-1)*a[n-2], makelist(a[n], n, 0, 20)); /* Franck Maminirina Ramaharo, Nov 27 2018 */
  • PARI
    my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-4*x)))) \\ Michel Marcus, Nov 25 2018

Formula

a(n) = Sum_{k=0..n} 4^(n - k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (8*n - 7)*a(n-1) - 16*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(sqrt(n) - 1/8) * 2^(2*n - 3/2) / (sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018

A321849 Expansion of e.g.f.: exp(x/(1-6*x)).

Original entry on oeis.org

1, 1, 13, 253, 6553, 211801, 8201701, 369979093, 19047250993, 1101705494833, 70715424362941, 4987040544656941, 383243311962126793, 31871863566298601353, 2851588139929576342933, 273093945561709965890821, 27871997808801906673665121
Offset: 0

Views

Author

Ludovic Schwob, Nov 19 2018

Keywords

Comments

For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.

Links

Crossrefs

Cf. A000262, A025168, A321837, A321847, A321848, A321850.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1-6*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Dec 08 2018
  • Magma
    [1] cat [&+[6^(n-k)* Factorial(n)/Factorial(k)*Binomial(n-1, k-1):  k in [0..n]]: n in [1.. 18]]; // Vincenzo Librandi, Dec 08 2018
  • Maple
    seq(coeff(series(factorial(n)*exp(x/(1-6*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018
  • Mathematica
    a[n_] := Sum[6^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (12n - 11)*a[n - 1] - 36(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)
  • PARI
    my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-6*x)))) \\ Michel Marcus, Nov 25 2018
  • Sage
    f= exp(x/(1-6*x))
g=f.taylor(x,0,13)
L=g.coefficients()
coeffs={c[1]:c[0]*factorial(c[1]) for c in L}
coeffs  # G. C. Greubel, Dec 08 2018

Formula

a(n) = Sum_{k=0..n} 6^(n-k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (12*n-11)*a(n-1) - 36*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(sqrt(2*n/3) - 1/12) * 6^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018

A321850 E.g.f.: exp(x/(1-7*x)).

Original entry on oeis.org

1, 1, 15, 337, 10081, 376461, 16849351, 878797165, 52324954977, 3501300491641, 260062721279551, 21228108532279881, 1888618754806601665, 181873163575529411077, 18846187172580219099831, 2090754000231731874682021, 247221828043044971020645441
Offset: 0

Views

Author

Ludovic Schwob, Nov 19 2018

Keywords

Comments

For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.

Links

Crossrefs

Cf. A000262, A025168, A321837, A321847, A321848, A321849.

Programs

  • Magma
    [1] cat [&+[7^(n-k)*Factorial(n)/Factorial(k)*Binomial(n-1, k-1):  k in [0..n]]: n in [1.. 18]]; // Vincenzo Librandi, Dec 08 2018
  • Maple
    seq(coeff(series(factorial(n)*exp(x/(1-7*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018
  • Mathematica
    a[n_] := Sum[7^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (14n - 13)*a[n - 1] - 49(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)
  • PARI
    my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-7*x)))) \\ Michel Marcus, Nov 25 2018

Formula

a(n) = Sum_{k=0..n} 7^(n-k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (14*n-13)*a(n-1) - 49*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(2*sqrt(n/7) - 1/14) * 7^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018

A341033 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-k*x)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 37, 73, 1, 1, 1, 9, 73, 361, 501, 1, 1, 1, 11, 121, 1009, 4361, 4051, 1, 1, 1, 13, 181, 2161, 17341, 62701, 37633, 1, 1, 1, 15, 253, 3961, 48081, 355951, 1044205, 394353, 1
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2021

Keywords

Examples

			Square array begins:
  1,   1,    1,     1,     1,      1, ...
  1,   1,    1,     1,     1,      1, ...
  1,   3,    5,     7,     9,     11, ...
  1,  13,   37,    73,   121,    181, ...
  1,  73,  361,  1009,  2161,   3961, ...
  1, 501, 4361, 17341, 48081, 108101, ...

Links

Crossrefs

Columns 0..7 give A000012, A000262, A025168, A321837, A321847, A321848, A321849, A321850.
Main diagonal gives A293146.
Cf. A253286, A341014.

Programs

  • Mathematica
    T[0, k_] = 1; T[n_, k_] := n!*Sum[If[k == n - j == 0, 1, k^(n - j)]*Binomial[n - 1, j - 1]/j!, {j, 1, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 03 2021 *)
  • PARI
    {T(n, k) = if(n==0, 1, n!*sum(j=1, n, k^(n-j)*binomial(n-1, j-1)/j!))}
  • PARI
    {T(n, k) = if(n<2, 1, (2*k*n-2*k+1)*T(n-1, k)-k^2*(n-1)*(n-2)*T(n-2, k))}

Formula

T(n,k) = Sum_{j=1..n} k^(n-j) * (n!/j!) * binomial(n-1,j-1) for n > 0.
T(n,k) = (2*k*n-2*k+1) * T(n-1,k) - k^2 * (n-1) * (n-2) * T(n-2,k) for n > 1.
Showing 1-5 of 5 results.

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