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.

A293720 Expansion of e.g.f.: exp(x + 4*x^2).

Original entry on oeis.org

1, 1, 9, 25, 241, 1041, 10681, 60649, 658785, 4540321, 51972841, 415198521, 4988808529, 44847866545, 563683953561, 5586645006601, 73228719433921, 788319280278849, 10747425123292105, 124265401483446361, 1757874020223846321, 21640338257575264081
Offset: 0

Views

Author

Seiichi Manyama, Oct 15 2017

Keywords

Links

Crossrefs

Column k=2 of A293724.
Column k=8 of A359762.
Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), this sequence (q=4).

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x+4*x^2) ))); // G. C. Greubel, Jul 12 2024
  • Mathematica
    CoefficientList[Series[E^(x + 4*x^2), {x,0,30}], x] * Range[0,30]! (* Vaclav Kotesovec, Oct 15 2017 *)
  • PARI
    my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(x+4*x^2)))
  • SageMath
    [(-2*i)^n*hermite(n, i/4) for n in range(31)] # G. C. Greubel, Jul 12 2024

Formula

a(n) ~ 2^((3*n-1)/2) * exp(-1/32 + sqrt(2*n)/4 - n/2) * n^(n/2). - Vaclav Kotesovec, Oct 15 2017
a(n) = (-2*i)^n * Hermite(n, i/4). - G. C. Greubel, Jul 12 2024

A293669 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 7, 1, 1, 1, 3, 13, 25, 1, 1, 1, 3, 13, 49, 81, 1, 1, 1, 3, 13, 73, 261, 331, 1, 1, 1, 3, 13, 73, 381, 1531, 1303, 1, 1, 1, 3, 13, 73, 501, 2611, 9073, 5937, 1, 1, 1, 3, 13, 73, 501, 3331, 19993, 63393, 26785, 1, 1, 1, 3, 13, 73, 501, 4051, 27553, 165873, 465769, 133651, 1
Offset: 0

Views

Author

Seiichi Manyama, Oct 14 2017

Keywords

Examples

			Square array begins:
   1,  1,   1,   1,   1, ...
   1,  1,   1,   1,   1, ...
   1,  3,   3,   3,   3, ...
   1,  7,  13,  13,  13, ...
   1, 25,  49,  73,  73, ...
   1, 81, 261, 381, 501, ...

Links

Crossrefs

Columns k=1..9 give A000012, A047974, A118589, A193930, A193931, A193932, A193933, A306623, A306624.
Rows n=0-1 give A000012.
Main diagonal gives A000262.
Cf. A293718, A293724.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(n-j, k)*binomial(n-1, j-1)*j!, j=1..min(n, k)))
    end:
seq(seq(A(n, 1+d-n), n=0..d), d=0..12);  # Alois P. Heinz, Nov 11 2020
  • Mathematica
    A[0, ] = 1; A[n /; n >= 0, k_ /; k >= 1] := A[n, k] = (n-1)!*Sum[j*A[n-j, k]/(n-j)!, {j, 1, Min[k, n]}]; A[, ] = 0;
Table[A[n, d-n+1], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 13 2021 *)

Formula

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(k,n)} j*A(n-j,k)/(n-j)!.

A293718 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} j*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 13, 1, 1, 1, 5, 31, 73, 1, 1, 1, 5, 31, 145, 281, 1, 1, 1, 5, 31, 241, 1181, 1741, 1, 1, 1, 5, 31, 241, 1661, 9661, 8485, 1, 1, 1, 5, 31, 241, 2261, 16861, 77155, 57233, 1, 1, 1, 5, 31, 241, 2261, 20461, 181315, 794081
Offset: 0

Views

Author

Seiichi Manyama, Oct 15 2017

Keywords

Examples

			Square array begins:
   1,   1,    1,    1,    1, ...
   1,   1,    1,    1,    1, ...
   1,   5,    5,    5,    5, ...
   1,  13,   31,   31,   31, ...
   1,  73,  145,  241,  241, ...
   1, 281, 1181, 1661, 2261, ...

Links

Crossrefs

Columns k=1..4 give A000012, A115329, A293716, A293717.
Rows n=0-1 give A000012.
Main diagonal gives A082579.
Cf. A293669, A293724.

Formula

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(k,n)} j^2*A(n-j,k)/(n-j)!.

A293721 E.g.f.: exp(x + 4*x^2 + 9*x^3).

Original entry on oeis.org

1, 1, 9, 79, 457, 5901, 66841, 720259, 11155089, 158315257, 2361665161, 42133751991, 720156599449, 13181971424389, 265545621153177, 5280775950377131, 111888028465044001, 2508562975185903729, 56507353426001537929, 1342159313030965211167
Offset: 0

Views

Author

Seiichi Manyama, Oct 15 2017

Keywords

Links

Crossrefs

Column k=3 of A293724.

Programs

  • Mathematica
    CoefficientList[Series[E^(x + 4*x^2 + 9*x^3), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 15 2017 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x+4*x^2+9*x^3)))

Formula

a(n) ~ 3^(n - 1/2) * n^(2*n/3) * exp(-392/6561 - 2*n/3 + 49*n^(1/3)/243 + 4*n^(2/3)/9). - Vaclav Kotesovec, Oct 15 2017

A293723 E.g.f.: exp(x + 4*x^2 + 9*x^3 + 16*x^4).

Original entry on oeis.org

1, 1, 9, 79, 841, 7821, 118681, 1782019, 28600209, 490277881, 9841912201, 200678086071, 4335000908569, 100095347642629, 2477069706227481, 63039923265570091, 1685240267891749921, 47330265699309738609, 1386300580285054314889
Offset: 0

Views

Author

Seiichi Manyama, Oct 15 2017

Keywords

Links

Crossrefs

Column k=4 of A293724.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[x+4x^2+9x^3+16x^4],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 04 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x+4*x^2+9*x^3+16*x^4)))
Showing 1-5 of 5 results.

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