A008299 -id:A008299 - cp's OEIS frontend

A076126 Triangle T(n,k) of associated Lah numbers, n>=2, k=1..floor(n/2).

2, 6, 24, 12, 120, 120, 720, 1080, 120, 5040, 10080, 2520, 40320, 100800, 40320, 1680, 362880, 1088640, 604800, 60480, 3628800, 12700800, 9072000, 1512000, 30240, 39916800, 159667200, 139708800, 33264000, 1663200, 479001600

Offset: 2

Vladeta Jovovic, Vladimir Baltic, Oct 31 2002

Number of partitions of {1,..,n} into k lists of size >1, where a list means an ordered subset, cf. A008297.

			2; 6; 24, 12; 120,120; 720,1080,120; 5040,10080, 2520; ...

Row sums give A052845, A008306, A008299.

T(n, k) = n!/k!*binomial(n-k-1, k-1), n>=2, k=1..floor(n/2). G.f.: G.f.: Sum_{n>=2, k=1..floor(n/2)} T(n, k)*x^n*y^k/n! = exp(x^2*y/(1-x))-1.

A174859 A triangle sequence of polynomial coefficients:p(x,n)=Sum[Binomial[n, k](-x)^kSum[StirlingS2[n, m]*x^m, {m, 0, n - k}], {k, 0, n}].

Original entry on oeis.org

1, 0, 1, 0, 1, -1, 0, 1, 0, -5, 0, 1, 3, -16, 15, 0, 1, 10, -40, 25, 56, 0, 1, 25, -81, -30, 370, -455, 0, 1, 56, -119, -469, 1841, -1960, -237, 0, 1, 119, -22, -2527, 7448, -5768, -7420, 16947, 0, 1, 246, 766, -10359, 24627, -2289, -76692, 126504, -64220, 0, 1

Offset: 0

Roger L. Bagula, Mar 31 2010

Row sums are:

{1, 1, 0, -4, 3, 52, -170, -887, 8778, -1416, -415734,...}.

			{1},
{0, 1},
{0, 1, -1},
{0, 1, 0, -5},
{0, 1, 3, -16, 15},
{0, 1, 10, -40, 25, 56},
{0, 1, 25, -81, -30, 370, -455},
{0, 1, 56, -119, -469, 1841, -1960, -237},
{0, 1, 119, -22, -2527, 7448, -5768, -7420, 16947},
{0, 1, 246, 766, -10359, 24627, -2289, -76692, 126504, -64220},
{0, 1, 501, 4265, -36320, 60215, 119760, -570627, 784245, -248280, -529494}

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 77.

Cf. A008299

Clear[p, x, n];
p[x_, n_] = Sum[Binomial[n, k]*(-x)^k*Sum[StirlingS2[n, m]*x^m, {m, 0, n - k}], {k, 0, n}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

p(x,n)=Sum[Binomial[n, k]*(-x)^k*Sum[StirlingS2[n, m]*x^m, {m, 0, n - k}], {k, 0, n}];

t(n,m)=coefficients(p(x,n))

A281269 Triangle read by rows: T(n,k) is the number of edge covers of the complete labeled graph on n nodes that are minimal and have exactly k edges, n>=2, 1<=k<=n-1.

Original entry on oeis.org

1, 0, 3, 0, 3, 4, 0, 0, 30, 5, 0, 0, 15, 150, 6, 0, 0, 0, 315, 525, 7, 0, 0, 0, 105, 3360, 1568, 8, 0, 0, 0, 0, 3780, 24570, 4284, 9, 0, 0, 0, 0, 945, 69300, 142380, 11070, 10, 0, 0, 0, 0, 0, 51975, 866250, 713790, 27555, 11, 0, 0, 0, 0, 0, 10395, 1455300, 8399160, 3250500, 66792, 12

Offset: 2

Geoffrey Critzer, Apr 25 2017

A minimal edge cover is an edge cover such that the removal of any edge in the cover destroys the covering property.

			1;
0, 3;
0, 3,  4;
0, 0, 30,   5;
0, 0, 15, 150,    6;
0, 0,  0, 315,  525,     7;
0, 0,  0, 105, 3360,  1568,      8;
0, 0,  0,   0, 3780, 24570,   4284,     9;
0, 0,  0,   0,  945, 69300, 142380, 11070, 10;

Eric Weisstein's World of Mathematics, Complete Graph
Eric Weisstein's World of Mathematics, Edge Cover
Eric Weisstein's World of Mathematics, Edge Cover Polynomial

Row sums give A053530.
First positive term in each even row is A001147.
First positive term in each odd row is A200142.

nn = 12; list = Range[0, nn]! CoefficientList[Series[Exp[z (Exp[x] - x - 1)], {x, 0, nn}], x];Table[Map[Drop[#, 1] &,
    Drop[Range[0, nn]! CoefficientList[Series[Exp[u z^2/2!] Sum[(u z)^j/j!*list[[j + 1]], {j, 0, nn}], {z, 0, nn}], {z, u}], 2]][[n, 1 ;; n]], {n, 1, nn - 1}] // Grid

E.g.f.: exp(y*x^2/2) * Sum_{j>=0} (y*x)^j/j! * Sum_{k=0..floor(j/2)} A008299(j,k)*x^k.

cp's OEIS Frontend

A076126 Triangle T(n,k) of associated Lah numbers, n>=2, k=1..floor(n/2).

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A174859 A triangle sequence of polynomial coefficients:p(x,n)=Sum[Binomial[n, k](-x)^kSum[StirlingS2[n, m]*x^m, {m, 0, n - k}], {k, 0, n}].

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A281269 Triangle read by rows: T(n,k) is the number of edge covers of the complete labeled graph on n nodes that are minimal and have exactly k edges, n>=2, 1<=k<=n-1.

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cp's OEIS Frontend

A076126 Triangle T(n,k) of associated Lah numbers, n>=2, k=1..floor(n/2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A174859 A triangle sequence of polynomial coefficients:p(x,n)=Sum[Binomial[n, k]*(-x)^k*Sum[StirlingS2[n, m]*x^m, {m, 0, n - k}], {k, 0, n}].

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Formula

A281269 Triangle read by rows: T(n,k) is the number of edge covers of the complete labeled graph on n nodes that are minimal and have exactly k edges, n>=2, 1<=k<=n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A174859 A triangle sequence of polynomial coefficients:p(x,n)=Sum[Binomial[n, k](-x)^kSum[StirlingS2[n, m]*x^m, {m, 0, n - k}], {k, 0, n}].