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.

Previous Showing 11-14 of 14 results.

A154961 2nd column of A154960.

Original entry on oeis.org

0, 1, 3, 25, 340, 7026, 204862, 8007602, 404077632, 25569505628, 1982619985192, 184861494417920, 20406183592852460, 2631875641089358912, 392163247878318070876, 66855512799464487146588, 12929525365915201064027856, 2815456378791384288128303192
Offset: 1

Views

Author

Mats Granvik, Jan 18 2009

Keywords

Comments

A052882 might have similarities with this sequence because A052882 is the 2nd column in table A154921 which is similar to A154960.

Crossrefs

Cf. A052882, A154921, A154960.

Programs

  • PARI
    { (matrix(30,30,i,j,(-1)^(i!=j)*stirling(i,j,2))^(-1))[,2] } \\ Max Alekseyev, Jun 17 2011

Extensions

More terms from Max Alekseyev, Jun 17 2011

A162498 Triangle read by rows: T(n, k) = Sum_{j=0..k} (-1)^j*binomial(n, j)*(k + 1 - j)^(n - 1).

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 23, 33, 11, 1, 425, 620, 220, 26, 1, 18129, 26525, 9520, 1180, 57, 1, 1721419, 2519664, 905765, 113050, 5649, 120, 1, 353654167, 517670461, 186123259, 23248085, 1166221, 25347, 247, 1, 153923102577, 225309742552, 81009042744, 10119247684, 507795498, 11059468, 109386, 502, 1
Offset: 1

Views

Author

Roger L. Bagula and Mats Granvik, Dec 06 2009

Keywords

Examples

			Triangle begins:
  {1},
  {1, 1},
  {3, 4, 1},
  {23, 33, 11, 1},
  {425, 620, 220, 26, 1},
  {18129, 26525, 9520, 1180, 57, 1},
  {1721419, 2519664, 905765, 113050, 5649, 120, 1},
  {353654167, 517670461, 186123259, 23248085, 1166221, 25347, 247, 1},
  {153923102577, 225309742552, 81009042744, 10119247684, 507795498, 11059468, 109386, 502, 1},
  ...

Crossrefs

Cf. A154921. An unsigned version of A055325.

Programs

  • Mathematica
    t[n_,k_]:=Sum[(-1)^j Binomial[n,j](k+1-j)^(n-1),{j,0,k}];
M[n_]:=Table[If[k <= m,(-1)^(m+k)*t[m,k],0],{k,0,n-2},{m,2,n}];
Flatten[Table[Table[Inverse[M[12]][[m,n]],{m,1,n}],{n,1,11}]]

A350531 Triangle read by rows: T(n,k) is the number of labeled loop-threshold graphs on vertex set [n] with k components, for n >= 1 and 1 <= k <= n.

Original entry on oeis.org

2, 3, 3, 13, 9, 4, 75, 52, 18, 5, 541, 375, 130, 30, 6, 4683, 3246, 1125, 260, 45, 7, 47293, 32781, 11361, 2625, 455, 63, 8, 545835, 378344, 131124, 30296, 5250, 728, 84, 9, 7087261, 4912515, 1702548, 393372, 68166, 9450, 1092, 108, 10
Offset: 1

Views

Author

David Galvin, Jan 03 2022

Keywords

Comments

Loop-threshold graphs are constructed from either a single unlooped vertex or a single looped vertex by iteratively adding isolated vertices (adjacent to nothing previously added) and looped dominating vertices (looped, and adjacent to everything previously added).

Examples

			Triangle begins:
        2;
        3,       3;
       13,       9,       4;
       75,      52,      18,      5;
      541,     375,     130,     30,     6;
     4683,    3246,    1125,    260,    45,    7;
    47293,   32781,   11361,   2625,   455,   63,    8;
   545835,  378344,  131124,  30296,  5250,  728,   84,   9;
  7087261, 4912515, 1702548, 393372, 68166, 9450, 1092, 108, 10;
  ...

Links

Crossrefs

Row sums are A000629.
Except at n = 1, first column is A000670.
Essentially the same as A154921 --- in A350531 (this triangle), replace the last nonzero entry in row m (this entry is m+1) with the two entries m, 1 to get A154921.
Cf. A173018.

Programs

  • Mathematica
    eulerian[n_, m_] := eulerian[n, m] =
  Sum[((-1)^k)*Binomial[n+1, k]*((m+1-k)^n), {k, 0, m+1}] (* eulerian[n, m] is an Eulerian number, counting the permutations of [n] with m descents *); T[1, 1] = 2; T[n_, 1] := T[n, 1] = Sum[eulerian[n, k]*(2^k), {k, 0, n - 1}]; T[n_, n_] := T[n, n] = n + 1; T[n_, k_] := T[n, k] = Binomial[n, k - 1]*T[n - k + 1, 1]; Table[T[n, k], {n, 1, 12}, {k, 1, n}]

Formula

T(n,n) = n+1 for n >= 1; T(n,1) = Sum_{j=0..n-1} A173018(n,j)*2^j for n >= 2; T(n,k) = binomial(n, k-1)*T(n-k+1,1) for n >= 3, 2 <= k <= n-1.

A171273 Matrix inverse of A060187.

Original entry on oeis.org

1, 1, 1, 5, 6, 1, 93, 115, 23, 1, 5993, 7436, 1518, 76, 1, 1272089, 1578757, 322762, 16330, 237, 1, 857402029, 1064110290, 217560951, 11012540, 160571, 722, 1, 1792650585525, 2224835452407, 454875884137, 23025275075, 335768223, 1512581, 2179, 1
Offset: 1

Views

Author

Roger L. Bagula and Mats Granvik, Dec 06 2009

Keywords

Examples

			{1},
{1, 1},
{5, 6, 1},
{93, 115, 23, 1},
{5993, 7436, 1518, 76, 1},
{1272089, 1578757, 322762, 16330, 237, 1},
{857402029, 1064110290, 217560951, 11012540, 160571, 722, 1},
{1792650585525, 2224835452407, 454875884137, 23025275075, 335768223, 1512581, 2179, 1},
{11464255554367057, 14228139328931096, 2908996087466828, 147249943814184, 2147290464886, 9673492136, 13945196, 6552, 1},
{222406320165016449457, 276025608122908733321, 56434463826320585284, 2856645864675796564, 41657391444153086, 187665608020478, 270538484020, 127141156, 19673, 1},
{13026233415367869864109781, 16166689855580307839632286, 3305339964838291288943901, 167312402773377971746920, 2439853795947184617546, 10991486289326969076, 15845312257310658, 7446608913000, 1152338433, 59038, 1}

Crossrefs

Cf. A060187, A055325, A154921,

Programs

  • Mathematica
    m = 2;
A[n_, 1] := 1
A[n_, n_] := 1
A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k]
a = Table[A[n, k], {n, 12}, {k, n}]
M[n_] := Table[If[k <= m, (-1)^(m + k)*a[[m, k]], 0], {k, 1, n}, {m, 1, n}]
Table[Table[Inverse[M[12]][[m, n]], {m, 1, n}], {n, 1, 11}]
Flatten[%]

Formula

A(n,k) = (m*n - m*k + 1) * A(n - 1, k - 1) + (m*k - (m - 1)) * A(n - 1, k), A(n,1) = A(n,n) = 1.
Previous Showing 11-14 of 14 results.

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