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 1-6 of 6 results.

A293724 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^2*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 9, 25, 1, 1, 1, 9, 79, 241, 1, 1, 1, 9, 79, 457, 1041, 1, 1, 1, 9, 79, 841, 5901, 10681, 1, 1, 1, 9, 79, 841, 7821, 66841, 60649, 1, 1, 1, 9, 79, 841, 10821, 118681, 720259, 658785, 1, 1, 1, 9, 79, 841, 10821, 136681, 1782019
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,    9,    9,    9,     9, ...
   1,   25,   79,   79,    79, ...
   1,  241,  457,  841,   841, ...
   1, 1041, 5901, 7821, 10821, ...

Links

Crossrefs

Columns k=1..4 give A000012, A293720, A293721, A293723.
Rows n=0-1 give A000012.
Main diagonal gives A255807.
Cf. A293669, A293718.

Formula

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(k,n)} j^3*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)!.

A334561 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, -1, -1, 1, -1, -1, 5, 1, 1, -1, -1, -1, 1, -1, 1, -1, -1, -1, 25, -41, 1, 1, -1, -1, -1, 1, 19, 31, -1, 1, -1, -1, -1, 1, 139, -209, 461, 1, 1, -1, -1, -1, 1, 19, 151, -2269, -895, -1, 1, -1, -1, -1, 1, 19, 871, -1429, 2801, -6481, 1, 1, -1, -1, -1, 1, 19, 151, 1091, -19039, 68615, 22591, -1
Offset: 0

Views

Author

Seiichi Manyama, May 06 2020

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,   5,   -1,  -1,  -1,  -1,  -1, ...
   1,   1,   25,   1,   1,   1,   1, ...
  -1, -41,   19, 139,  19,  19,  19, ...
   1,  31, -209, 151, 871, 151, 151, ...

Links

Crossrefs

Columns k=1..5 give A033999, A000321, A334562, A334564, A334565.
Main diagonal gives A293116.
Cf. A293669, A334568.

Formula

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

A193933 E.g.f. A(x) = exp(x+x^2+x^3+x^4+x^5+x^6+x^7).

Original entry on oeis.org

1, 1, 3, 13, 73, 501, 4051, 37633, 354033, 3870793, 46240291, 597877941, 8298856633, 122751616573, 1921371570483, 31604885804521, 552755907700321, 10156326950561553, 195421314725788483, 3926668816722630493, 82199760488718697641, 1789438454541407131141
Offset: 0

Views

Author

Vladimir Kruchinin, Aug 09 2011

Keywords

Links

Crossrefs

Column k=7 of A293669.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=1..min(n, 7)))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 29 2017
  • Mathematica
    terms = 22;
CoefficientList[E^Total[x^Range[7]] + O[x]^terms, x] Range[0, terms-1]! (* Jean-François Alcover, Nov 11 2020 *)
  • Maxima
    a(n):=if n=0 then 1 else n!*sum(sum((-1)^i*binomial(k, k-i)*binomial(n-7*i-1, k-1), i, 0, (n-k)/7)/k!, k, 1, n);
makelist(a(n),n,0,20);

Formula

E.g.f.: exp(Sum_{j=1..7} x^j).
a(n) = n!*sum(k=1..n, sum(i=0..(n-k)/7, (-1)^i*binomial(k,k-i)*binomial(n-7*i-1,k-1))/k!), n>0, a(0)=1.

A306623 Expansion of e.g.f. exp(Sum_{k=1..8} x^k).

Original entry on oeis.org

1, 1, 3, 13, 73, 501, 4051, 37633, 394353, 4233673, 51683491, 684364341, 9755819833, 148749428413, 2411870539443, 41369113878121, 746931540551521, 14128241450715153, 281805883597349443, 5880463849670410333, 128050607992266620841, 2903233047048502113541
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2019

Keywords

Links

Crossrefs

Column 8 of A293669.

Programs

  • Mathematica
    m=21; CoefficientList[Series[Exp[Sum[x^k, {k,1,8}]], {x, 0, m}], x] * Range[0,m]! (* Amiram Eldar, Mar 01 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, 8, x^k))))

Formula

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..min(8,n)} k*a(n-k)/(n-k)! for n > 0.

A306624 Expansion of e.g.f. exp(Sum_{k=1..9} x^k).

Original entry on oeis.org

1, 1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553, 55312291, 744239541, 10793656633, 167689950013, 2775839905203, 48726598412521, 903159189729121, 17607070923233553, 359702718305842243, 7673827033741108573, 171586828999546057641, 3999150173195168500741
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2019

Keywords

Links

Crossrefs

Column 9 of A293669.

Programs

  • Mathematica
    m=21; CoefficientList[Series[Exp[Sum[x^k, {k,1,9}]], {x, 0, m}], x] * Range[0,m]! (* Amiram Eldar, Mar 01 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, 9, x^k))))

Formula

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..min(9,n)} k*a(n-k)/(n-k)! for n > 0.
Showing 1-6 of 6 results.

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