A151817 -id:A151817 - cp's OEIS frontend

A155951 Triangle read by rows. Let q(x,n) = -((x - 1)^(2n + 1)/x^n)Sum[(k + n)^nBinomial[k, n]x^k, {k, 0, Infinity}]; p(x,n)=q(x,n)+x^n*q(1/x,n); then row n gives coefficients of p(x,n).

2, 4, 17, -10, 17, 208, -88, -88, 208, 4177, -4708, 4422, -4708, 4177, 98976, -123888, 55152, 55152, -123888, 98976, 3001609, -5204582, 5360567, -4984628, 5360567, -5204582, 3001609, 105133568, -210753520, 208361232, -85444000, -85444000

Offset: 0

Roger L. Bagula, Jan 31 2009

Row sums are in A151817.

			{2},
{4},
{17, -10, 17},
{208, -88, -88, 208},
{4177, -4708, 4422, -4708, 4177},
{98976, -123888, 55152, 55152, -123888, 98976},
{3001609, -5204582, 5360567, -4984628, 5360567, -5204582, 3001609},
{105133568, -210753520, 208361232, -85444000, -85444000, 208361232, -210753520, 105133568},
{4300732097, -10315512136, 13267499516, -12384821752, 11302041350, -12384821752, 13267499516, -10315512136, 4300732097},
{198225072640, -539802938440, 752937755480, -641425101400, 247708437320, 247708437320, -641425101400, 752937755480, -539802938440, 198225072640},
{10243486784401, -31622720552146, 50805231998853, -55277019174408, 48150459465066, -43257991897932, 48150459465066, -55277019174408, 50805231998853, -31622720552146, 10243486784401}

Clear[p, x, n, m];
p[x_, n_] = -((x - 1)^(2*n + 1)/x^n)*Sum[(k + n)^n*Binomial[k, n]*x^k, {k, 0, Infinity}];
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]
+ Reverse[ CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]], {n, 0, 10}];
Flatten[%]

q(x,n)=-((x - 1)^(2*n + 1)/x^n)*Sum[(k + n)^n*Binomial[k, n]*x^k, {k, 0, Infinity}];
p(x,n)=q(x,n)+x^n*q(1/x,n);
t(n,m)=coefficients(p(x,n))

Edited by N. J. A. Sloane, Jul 05 2009

A370208 Triangular array read by rows. T(n,k) is the number of idempotent binary relations on [n] having no proper power primitive (A360718) with exactly k irreflexive points.

Original entry on oeis.org

1, 1, 1, 3, 6, 13, 39, 87, 348, 24, 841, 4205, 480, 11643, 69858, 9420, 240, 227893, 1595251, 206640, 9240, 6285807, 50286456, 5389552, 299040, 3360, 243593041, 2192337369, 172041408, 9848160, 211680

Offset: 0

Geoffrey Critzer, Feb 11 2024

			 Triangle begins
      1;
      1,       1;
      3,       6;
     13,      39;
     87,     348,     24;
    841,    4205,    480;
  11643,   69858,   9420,  240;
 227893, 1595251, 206640, 9240;
 ...

David Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research, National Bureau of Standards, Vol 67B No. 4 Oct-Dec 1963.

Cf. A360718 (row sums), A001831 (column k=0), A360743 (T(n,0) + T(n,1) ), A151817 (T(2n,n) for n>=2), A002031.

nn = 9; A[x_] := Sum[x^n/n! Exp[(2^n - 1) x], {n, 0, nn}];
c[x_] := Log[A[x]] - x; Map[Select[#, # > 0 &] &,
 Range[0, nn]! CoefficientList[
   Series[2 (Exp[ y x D[c[ x], x]/2] - 1) Exp[c[x]] Exp[ x] +
     Exp[c[ x]] (y x Exp[  x] + Exp[ x]), {x, 0, nn}], {x, y}]]

E.g.f.: 2(exp(y*x*c'(x)/2)-1)*exp(c(x))*exp(x) + exp(c(x))*(y*x*exp(x) + exp(x)) where c(x) is the e.g.f. for A002031.

cp's OEIS Frontend

A155951 Triangle read by rows. Let q(x,n) = -((x - 1)^(2n + 1)/x^n)Sum[(k + n)^nBinomial[k, n]x^k, {k, 0, Infinity}]; p(x,n)=q(x,n)+x^n*q(1/x,n); then row n gives coefficients of p(x,n).

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A370208 Triangular array read by rows. T(n,k) is the number of idempotent binary relations on [n] having no proper power primitive (A360718) with exactly k irreflexive points.

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cp's OEIS Frontend

A155951 Triangle read by rows. Let q(x,n) = -((x - 1)^(2*n + 1)/x^n)*Sum[(k + n)^n*Binomial[k, n]*x^k, {k, 0, Infinity}]; p(x,n)=q(x,n)+x^n*q(1/x,n); then row n gives coefficients of p(x,n).

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Extensions

A370208 Triangular array read by rows. T(n,k) is the number of idempotent binary relations on [n] having no proper power primitive (A360718) with exactly k irreflexive points.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A155951 Triangle read by rows. Let q(x,n) = -((x - 1)^(2n + 1)/x^n)Sum[(k + n)^nBinomial[k, n]x^k, {k, 0, Infinity}]; p(x,n)=q(x,n)+x^n*q(1/x,n); then row n gives coefficients of p(x,n).