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.

A294254 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} (1-x^j) - 1).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 1, 0, 1, -1, -1, -1, 0, 1, -1, -1, 11, 1, 0, 1, -1, -1, 5, -23, -1, 0, 1, -1, -1, 5, 25, -101, 1, 0, 1, -1, -1, 5, 1, -41, 991, -1, 0, 1, -1, -1, 5, 1, 199, -1769, -1849, 1, 0, 1, -1, -1, 5, 1, 79, -1409, 7181, -24751, -1, 0, 1, -1, -1, 5
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2017

Keywords

Examples

			Square array A(n,k) begins:
   1,  1,    1,   1,   1, ...
   0, -1,   -1,  -1,  -1, ...
   0,  1,   -1,  -1,  -1, ...
   0, -1,   11,   5,   5, ...
   0,  1,  -23,  25,   1, ...
   0, -1, -101, -41, 199, ...

Links

Crossrefs

Columns k=0..5 give A000007, A033999, A294255, A294256, A294257, A294258.
Rows n=0 gives A000012.
Main diagonal gives A294260.
Cf. A294250.

Formula

B(j,k) is the coefficient of Product_{i=1..k} (1-x^i).
A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(A000217(k),n)} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0.

A294212 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1-x^j) - 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 1, 5, 13, 0, 1, 1, 5, 25, 73, 0, 1, 1, 5, 31, 193, 501, 0, 1, 1, 5, 31, 241, 1601, 4051, 0, 1, 1, 5, 31, 265, 2261, 16741, 37633, 0, 1, 1, 5, 31, 265, 2501, 25501, 190345, 394353, 0, 1, 1, 5, 31, 265, 2621, 29461, 319915, 2509025
Offset: 0

Views

Author

Seiichi Manyama, Oct 25 2017

Keywords

Examples

			Square array B(j,k) begins:
   1,   1,    1,    1,    1, ...
   0,   1,    1,    1,    1, ...
   0,   1,    2,    2,    2, ...
   0,   1,    2,    3,    3, ...
   0,   1,    3,    4,    5, ...
   0,   1,    3,    5,    6, ...
Square array A(n,k) begins:
   1,   1,    1,    1,    1, ...
   0,   1,    1,    1,    1, ...
   0,   3,    5,    5,    5, ...
   0,  13,   25,   31,   31, ...
   0,  73,  193,  241,  265, ...
   0, 501, 1601, 2261, 2501, ...

Links

Crossrefs

Columns k=0..5 give A000007, A000262, A294213, A294214, A294215, A294216.
Rows n=0 gives A000012.
Main diagonal gives A058892.
Cf. A058398, A294250, A294254, A294289.

Formula

B(j,k) is the coefficient of Product_{i=1..k} 1/(1-x^i).
A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0.

A294253 E.g.f.: exp((1+x)*(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5) - 1).

Original entry on oeis.org

1, 1, 3, 19, 121, 1041, 9931, 106723, 1313649, 17830081, 265652371, 4259421651, 74011854313, 1374298028689, 27230459440731, 573414615381091, 12723857450638561, 297915550887491073, 7328943525355675939, 188820746254730967571, 5086439179764958688601
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2017

Keywords

Links

Crossrefs

Column k=5 of A294250.
Cf. A294216.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[Times@@(1+x^Range[5])-1],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 16 2023 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp((1+x)*(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5)-1)))

Formula

Recurrence: a(n) = a(n-1) + 2*(n-1)*a(n-2) + 6*(n-2)*(n-1)*a(n-3) + 8*(n-3)*(n-2)*(n-1)*a(n-4) + 15*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5) + 18*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6) + 21*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7) + 24*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-8) + 27*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-9) + 30*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n - 10) + 22*(n - 10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n - 11) + 24*(n - 11)*(n - 10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n - 12) + 13*(n - 12)*(n - 11)*(n - 10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n - 13) + 14*(n - 13)*(n - 12)*(n - 11)*(n - 10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n - 14) + 15*(n - 14)*(n - 13)*(n - 12)*(n - 11)*(n - 10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n - 15). - Vaclav Kotesovec, Dec 02 2021

A294251 E.g.f.: exp((1+x)*(1+x^2)*(1+x^3) - 1).

Original entry on oeis.org

1, 1, 3, 19, 97, 681, 5971, 50443, 512289, 5821777, 67968451, 880345731, 12230425153, 177123301369, 2759869352787, 45321758937211, 776008738668481, 14020831026652833, 264488668445049859, 5187662544182915827, 106241960755765408161, 2257485400533886011721
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2017

Keywords

Links

Crossrefs

Column k=3 of A294250.
Cf. A294214, A294291.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp((1+x)*(1+x^2)*(1+x^3)-1)))

Formula

a(n) = a(n-1) + 2*(n-1)*a(n-2) + 6*(n-2)*(n-1)*a(n-3) + 4*(n-3)*(n-2)*(n-1)*a(n-4) + 5*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5) + 6*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6). - Vaclav Kotesovec, Dec 02 2021

A294252 E.g.f.: exp((1+x)*(1+x^2)*(1+x^3)*(1+x^4) - 1).

Original entry on oeis.org

1, 1, 3, 19, 121, 921, 8491, 89083, 1004529, 12855601, 180798931, 2688972771, 43446800233, 753263746249, 13821916933371, 268513532534731, 5515341294313441, 119063602254649953, 2692597826408668579, 63786073308011409331, 1576811081112158751321
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2017

Keywords

Links

Crossrefs

Column k=4 of A294250.
Cf. A294215, A294292.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp((1+x)*(1+x^2)*(1+x^3)*(1+x^4)-1)))

Formula

a(n) = a(n-1) + 2*(n-1)*a(n-2) + 6*(n-2)*(n-1)*a(n-3) + 8*(n-3)*(n-2)*(n-1)*a(n-4) + 10*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5) + 12*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6) + 14*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7) + 8*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-8) + 9*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-9) + 10*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-10). - Vaclav Kotesovec, Dec 02 2021
Showing 1-5 of 5 results.

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