A014479 -id:A014479 - cp's OEIS frontend

A142707 Coefficients of derivatives of MacMahon polynomials (A060187): p(x,n)=2^n(1 - x)^(1 + n)LerchPhi[x, -n, 1/2]; p'(x,n)=(d/dx)p{x,n).

1, 6, 2, 23, 46, 3, 76, 460, 228, 4, 237, 3364, 5046, 948, 5, 722, 21086, 70644, 42172, 3610, 6, 2179, 121314, 779169, 1038892, 303285, 13074, 7, 6552, 663224, 7455864, 18700056, 12426440, 1989672, 45864, 8, 19673, 3512680, 65123916, 277653176

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 24 2008

Row sums are:A014479

0, 1, 8, 72, 768, 9600, 138240, 2257920, 41287680, 836075520, 18579456000.

			{1},
{6, 2},
{23, 46, 3},
{76, 460, 228, 4},
{237, 3364, 5046, 948, 5},
{722, 21086, 70644, 42172, 3610, 6},
{2179, 121314, 779169, 1038892, 303285, 13074, 7},
{6552, 663224, 7455864, 18700056, 12426440, 1989672, 45864, 8},
{19673, 3512680, 65123916, 277653176, 347066470, 130247832, 12294380, 157384, 9},
{59038, 18232282, 534902712, 3627693128, 7635462340, 5441539692, 1248106328, 72929128, 531342, 10}

Cf. A060187, A014479.

Clear[p, x, n, a]; p[x_, n_] = 2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2]; Table[FullSimplify[Expand[D[p[x, n], x]]], {n, 0, 10}]; Table[CoefficientList[FullSimplify[Expand[D[p[x, n], x]]], x], {n, 0, 10}]; Flatten[%]

p(x,n)=2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2]; p'(x,n)=(d/dx)p{x,n); t(n,m)=Coefficients(p'(x,n)).

A225474 Triangle read by rows, k!2^ks_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 2, 3, 8, 8, 15, 46, 72, 48, 105, 352, 688, 768, 384, 945, 3378, 7600, 11040, 9600, 3840, 10395, 39048, 97112, 167040, 193920, 138240, 46080, 135135, 528414, 1418648, 2754192, 3857280, 3736320, 2257920, 645120, 2027025, 8196480, 23393376, 49824768, 79892736

Offset: 0

Peter Luschny, May 19 2013

The Stirling-Frobenius cycle numbers are defined in A225470.

			[n\k][ 0,    1,    2,     3,    4,    5]
[0]    1,
[1]    1,    2,
[2]    3,    8,    8,
[3]   15,   46,   72,    48,
[4]  105,  352,  688,   768,  384,
[5]  945, 3378, 7600, 11040, 9600, 3840.

Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

Cf. A028338, A225479 (m=1), A048594, A161198, A225475.

Cf. A000079, A000142, A000165, A001147, A014479, A028338.

SFCSO[n_, k_, m_] := SFCSO[n, k, m] = If[k>n || k<0, 0, If[n == 0 && k == 0, 1, m*k*SFCSO[n-1, k-1, m] + (m*n-1)*SFCSO[n-1, k, m]]]; Table[SFCSO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 05 2014, translated from Sage *)

@CachedFunction
def SF_CSO(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return m*k*SF_CSO(n-1, k-1, m) + (m*n-1)*SF_CSO(n-1, k, m)
for n in (0..8): [SF_CSO(n, k, 2) for k in (0..n)]

For a recurrence see the Sage program.
T(n, 0) ~ A001147; T(n, n) ~ A000165; T(n, n-1) ~ A014479.
T(n,k) = A028338(n,k) * A000165(k) = A225475(n,k) * A000079(k) = A161198(n,k) * A000142(k). - Philippe Deléham, Jun 25 2015

A255908 Triangle read by rows: T(n,L) = number of rho-labeled graphs with n edges whose labeling is bipartite with boundary value L.

Original entry on oeis.org

2, 4, 8, 8, 32, 48, 16, 128, 288, 384, 32, 512, 1728, 3072, 3840, 64, 2048, 10368, 24576, 38400, 46080, 128, 8192, 62208, 196608, 384000, 552960, 645120, 256, 32768, 373248, 1572864, 3840000, 6635520, 9031680, 10321920, 512, 131072, 2239488, 12582912, 38400000, 79626240, 126443520, 165150720, 185794560, 1024, 524288, 13436928, 100663296, 384000000, 955514880, 1770209280, 2642411520, 3344302080, 3715891200

Offset: 1

Christian Barrientos and Sarah Minion, Mar 10 2015

A graph with n edges is rho-labeled if there exists a one-to-one mapping from its vertex set to {0,1,...,2n} such that every edge receives as label the absolute difference of its end-vertices and the edge labels are x_1, x_2, ..., x_n where x_i = i or x_i = 2n + 1 - i. A rho-labeling of a bipartite graph is said to be bipartite when the labels of one stable set are smaller than the labels on the other stable set. The largest of the smaller vertex labels is its boundary value.
From Robert G. Wilson v, Jul 05 2015: (Start)
The columns:
T(n, 0) = 2^n,
T(n, 1) = 2^(2n-1),
T(n, 2) = 2^(n+1)*3^(n-2),
T(n, 3) = 3*2^(3n-5),
T(n, 4) = 3*2^(n+3)*5^(n-4),
T(n, 5) = 5*2^(2n-2)*3^(n-4), etc.
The diagonals:
the main,            T(n, n-1) = 2^n*n*(n-1!) = 2*A002866,
the second diagonal, T(n, n-2) = 2^n*(n-1)^2*(n-2)! = 4*A014479,
the third diagonal,  T(n, n-3) = 2^n*(n-2)^3*(n-3)!,
the k_th diagonal,   T(n, n-k) = 2^n*(n-k)^k*(n-k)!, etc.
... (End)

			For n=5 and L=1, T(5,1)=(2^5)*(1!)*(1+1)^(5-1)=512.
For n=9 and L=3, T(9,3)=12582912.
Triangle, T, begins:
-----------------------------------------------------------------------------
n\L |   0       1         2          3          4          5           6
----|------------------------------------------------------------------------
1   |   2;
2   |   4,      8;
3   |   8,     32,       48;
4   |  16,    128,      288,       384;
5   |  32,    512,     1728,      3072,      3840;
6   |  64,   2048,    10368,     24576,     38400,     46080;
7   | 128,   8192,    62208,    196608,    384000,    552960,     645120;
8   | 256,  32768,   373248,   1572864,   3840000,   6635520,    9031680, ...
...
For n=2 and L=1, T(2,1)=8, because: the bipartite graph <{v1,v2,v3},{x1=v1v2,x2=v1v3}> has rho-labelings (2,1,3),(2,1,4) with L=1 on the stable set {x2} and rho-labelings (1,2,0),(0,4,1) with L=1 on the stable set {x1,x3}; the bipartite graph <{v1,v2,v3,v4},{x1=v1v2,x2=v3v4}> has rho-labeling (0,4,1,3),(1,2,0,3) with L=1 on the stable set {v1,v3} and rho-labeling (4,0,3,1),(2,1,3,0) with L=1 on the stable set {v2,v4}. - _Danny Rorabaugh_, Apr 03 2015

Joseph A. Gallian, A dynamic survey of graph labeling, Elec. J. Combin., (2014), #DS6.

[2^n*Factorial(l)*(l+1)^(n-l): l in [0..n-1], n in [1..10]]; // Bruno Berselli, Aug 05 2015

t[n_, l_] := 2^n*l!(l+1)^(n-l); Table[ t[n, l], {n, 8}, {l, 0, n-1}] // Flatten (* Robert G. Wilson v, Jul 05 2015 *)

For n>=1, 0<=L<=n-1, T(n,L)=(2^n)*(L!)*(L+1)^(n-L).

cp's OEIS Frontend

A142707 Coefficients of derivatives of MacMahon polynomials (A060187): p(x,n)=2^n(1 - x)^(1 + n)LerchPhi[x, -n, 1/2]; p'(x,n)=(d/dx)p{x,n).

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A225474 Triangle read by rows, k!2^ks_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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A255908 Triangle read by rows: T(n,L) = number of rho-labeled graphs with n edges whose labeling is bipartite with boundary value L.

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cp's OEIS Frontend

A142707 Coefficients of derivatives of MacMahon polynomials (A060187): p(x,n)=2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2]; p'(x,n)=(d/dx)p{x,n).

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A225474 Triangle read by rows, k!*2^k*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A255908 Triangle read by rows: T(n,L) = number of rho-labeled graphs with n edges whose labeling is bipartite with boundary value L.

Views

Author

Keywords

Comments

Examples

Links

Programs

Magma

Mathematica

Formula

A142707 Coefficients of derivatives of MacMahon polynomials (A060187): p(x,n)=2^n(1 - x)^(1 + n)LerchPhi[x, -n, 1/2]; p'(x,n)=(d/dx)p{x,n).

A225474 Triangle read by rows, k!2^ks_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.