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 1-8 of 8 results.

A362176 Expansion of e.g.f. exp(x * (1-2*x)).

Original entry on oeis.org

1, 1, -3, -11, 25, 201, -299, -5123, 3249, 167185, 50221, -6637179, -8846903, 309737689, 769776645, -16575533939, -62762132639, 998072039457, 5265897058909, -66595289781995, -466803466259079, 4860819716300521, 44072310882063157, -383679824152382691
Offset: 0

Views

Author

Seiichi Manyama, Apr 10 2023

Keywords

Links

Crossrefs

Column k=4 of A362277.
Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), this sequence (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x-2*x^2) ))); // G. C. Greubel, Jul 12 2024
  • Mathematica
    With[{m=30}, CoefficientList[Series[Exp[x-2*x^2], {x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Jul 12 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-2*x))))
  • SageMath
    [(-sqrt(2))^n*hermite(n, 1/(2*sqrt(2))) for n in range(31)] # G. C. Greubel, Jul 12 2024

Formula

a(n) = a(n-1) - 4*(n-1)*a(n-2) for n > 1.
a(n) = n! * Sum_{k=0..floor(n/2)} (-2)^k / (k! * (n-2*k)!).
a(n) = (-sqrt(2))^n * Hermite(n, 1/(2*sqrt(2))). - G. C. Greubel, Jul 12 2024

A362177 Expansion of e.g.f. exp(x * (1-3*x)).

Original entry on oeis.org

1, 1, -5, -17, 73, 481, -1709, -19025, 52753, 965953, -1882709, -59839889, 64418905, 4372890913, -651783677, -367974620369, -309314089439, 35016249465985, 66566286588763, -3715188655737617, -11303745326856599, 434518893361657441, 1858790804545588915
Offset: 0

Views

Author

Seiichi Manyama, Apr 10 2023

Keywords

Links

Crossrefs

Column k=6 of A362277.
Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), this sequence (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x-3*x^2) ))); // G. C. Greubel, Jul 12 2024
  • Mathematica
    With[{m=30}, CoefficientList[Series[Exp[x-3*x^2], {x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Jul 12 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-3*x))))
  • SageMath
    [(-sqrt(3))^n*hermite(n, 1/(2*sqrt(3))) for n in range(31)] # G. C. Greubel, Jul 12 2024

Formula

a(n) = a(n-1) - 6*(n-1)*a(n-2) for n > 1.
a(n) = n! * Sum_{k=0..floor(n/2)} (-3)^k / (k! * (n-2*k)!).
a(n) = (-sqrt(3))^n * Hermite(n, 1/(2*sqrt(3))). - G. C. Greubel, Jul 12 2024

A362302 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (-k/6)^j * binomial(n-2*j,j)/(n-2*j)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, -1, -3, 1, 1, 1, 1, -2, -7, -9, 1, 1, 1, 1, -3, -11, -19, -9, 1, 1, 1, 1, -4, -15, -29, 1, 36, 1, 1, 1, 1, -5, -19, -39, 31, 211, 225, 1, 1, 1, 1, -6, -23, -49, 81, 526, 1009, 477, 1, 1, 1, 1, -7, -27, -59, 151, 981, 2353, 953, -819, 1
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2023

Keywords

Examples

			Square array begins:
  1,  1,   1,   1,   1,   1,   1, ...
  1,  1,   1,   1,   1,   1,   1, ...
  1,  1,   1,   1,   1,   1,   1, ...
  1,  0,  -1,  -2,  -3,  -4,  -5, ...
  1, -3,  -7, -11, -15, -19, -23, ...
  1, -9, -19, -29, -39, -49, -59, ...
  1, -9,   1,  31,  81, 151, 241, ...

Links

Crossrefs

Columns k=0..2 give A000012, A351929, A362309.
Main diagonal gives A362303.
T(n,2*n) gives A362304.
T(n,6*n) gives A362305.
Cf. A362043, A362277.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\3, (-k/6)^j/(j!*(n-3*j)!));

Formula

E.g.f. of column k: exp(x - k*x^3/6).
T(n,k) = T(n-1,k) - k * binomial(n-1,2) * T(n-3,k) for n > 2.
T(n,k) = n! * Sum_{j=0..floor(n/3)} (-k/6)^j / (j! * (n-3*j)!).

A362276 a(n) = n! * Sum_{k=0..floor(n/2)} (-n/2)^k * binomial(n-k,k)/(n-k)!.

Original entry on oeis.org

1, 1, -1, -8, 25, 326, -1709, -31016, 228257, 5311900, -50337449, -1429574464, 16573668409, 555724876552, -7619288730325, -294582728145824, 4662562423032961, 204200579987319824, -3664348770051277073, -179294278761195862400, 3597007651803106610201
Offset: 0

Views

Author

Seiichi Manyama, Apr 13 2023

Keywords

Links

Crossrefs

Main diagonal of A362277.
Cf. A277614.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sqrt(lambertw(x^2)))/(1+lambertw(x^2))))

Formula

a(n) = n! * [x^n] exp(x - n*x^2/2).
E.g.f.: exp( sqrt( LambertW(x^2) ) ) / (1 + LambertW(x^2)).

A362282 a(n) = n! * Sum_{k=0..floor(n/2)} (-n)^k * binomial(n-k,k)/(n-k)!.

Original entry on oeis.org

1, 1, -3, -17, 145, 1401, -19619, -267833, 5214273, 91975825, -2292948899, -49586832129, 1506939887377, 38595456391753, -1383612408628995, -40951481342092649, 1691614670048805121, 56809502720559644577, -2656760323700732460227, -99810124102484722532465
Offset: 0

Views

Author

Seiichi Manyama, Apr 14 2023

Keywords

Links

Crossrefs

Cf. A362276, A362277, A362281.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sqrt(lambertw(2*x^2)/2))/(1+lambertw(2*x^2))))

Formula

a(n) = A362277(n,2*n).
a(n) = n! * [x^n] exp(x - n*x^2).
E.g.f.: exp( sqrt( LambertW(2*x^2)/2 ) ) / (1 + LambertW(2*x^2)).

A362394 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (-k/2)^j * (j+1)^(n-j-1) / (j! * (n-2*j)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -5, 1, 1, 1, -2, -11, -14, 1, 1, 1, -3, -17, -11, 56, 1, 1, 1, -4, -23, 10, 381, 736, 1, 1, 1, -5, -29, 49, 976, 2461, 1114, 1, 1, 1, -6, -35, 106, 1841, 3736, -21083, -45156, 1, 1, 1, -7, -41, 181, 2976, 3121, -106910, -449623, -428660, 1
Offset: 0

Views

Author

Seiichi Manyama, Apr 20 2023

Keywords

Examples

			Square array begins:
  1,   1,    1,    1,    1,    1,     1, ...
  1,   1,    1,    1,    1,    1,     1, ...
  1,   0,   -1,   -2,   -3,   -4,    -5, ...
  1,  -5,  -11,  -17,  -23,  -29,   -35, ...
  1, -14,  -11,   10,   49,  106,   181, ...
  1,  56,  381,  976, 1841, 2976,  4381, ...
  1, 736, 2461, 3736, 3121, -824, -9539, ...

Links

Crossrefs

Columns k=0..3 give A000012, A362395, A362396, A362397.
Cf. A362277, A362377.

Programs

  • PARI
    T(n, k) = n! * sum(j=0, n\2, (-k/2)^j*(j+1)^(n-j-1)/(j!*(n-2*j)!));

Formula

E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x - k*x^2/2 * A_k(x)).
A_k(x) = exp(x - LambertW(k*x^2/2 * exp(x))).
A_k(x) = 2 * LambertW(k*x^2/2 * exp(x))/(k*x^2) for k > 0.

A362278 Expansion of e.g.f. exp(x - 3*x^2/2).

Original entry on oeis.org

1, 1, -2, -8, 10, 106, -44, -1952, -1028, 45820, 73576, -1301024, -3729032, 43107832, 188540080, -1621988864, -10106292464, 67749173008, 583170088672, -3075285253760, -36315980308064, 148201134917536, 2436107894325568, -7345167010231808
Offset: 0

Views

Author

Seiichi Manyama, Apr 13 2023

Keywords

Links

Crossrefs

Column k=3 of A362277.
Cf. A115327.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x-3 x^2/2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 18 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-3*x^2/2)))

Formula

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

A362279 Expansion of e.g.f. exp(x - 5*x^2/2).

Original entry on oeis.org

1, 1, -4, -14, 46, 326, -824, -10604, 18236, 442396, -378224, -22498024, -1695704, 1348185736, 1458406496, -92914595024, -202295082224, 7230872519696, 24425954508736, -626352572263904, -2946818250593824, 59688438975796576, 369104355288148096
Offset: 0

Views

Author

Seiichi Manyama, Apr 13 2023

Keywords

Links

Crossrefs

Column k=5 of A362277.
Cf. A115331.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-5*x^2/2)))

Formula

a(n) = a(n-1) - 5*(n-1)*a(n-2) for n > 1.
a(n) = n! * Sum_{k=0..floor(n/2)} (-5/2)^k / (k! * (n-2*k)!).
Showing 1-8 of 8 results.

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