A176491
Triangle T(n,k) = binomial(n,k) + A176490(n,k) - 1 read along rows 0<=k<=n.
Original entry on oeis.org
1, 1, 1, 1, 10, 1, 1, 35, 35, 1, 1, 104, 300, 104, 1, 1, 297, 1992, 1992, 297, 1, 1, 846, 11747, 25982, 11747, 846, 1, 1, 2431, 64969, 275375, 275375, 64969, 2431, 1, 1, 7060, 346246, 2573576, 4831272, 2573576, 346246, 7060, 1, 1, 20693, 1804214, 22163246
Offset: 0
1;
1, 1;
1, 10, 1;
1, 35, 35, 1;
1, 104, 300, 104, 1;
1, 297, 1992, 1992, 297, 1;
1, 846, 11747, 25982, 11747, 846, 1;
1, 2431, 64969, 275375, 275375, 64969, 2431, 1;
1, 7060, 346246, 2573576, 4831272, 2573576, 346246, 7060, 1;
1, 20693, 1804214, 22163246, 70723772, 70723772, 22163246, 1804214, 20693, 1;
1, 61082, 9268821, 180504510, 916661604, 1542816966, 916661604, 180504510, 9268821, 61082, 1;
-
A176491 := proc(n,k)
A176490(n,k)+binomial(n,k)-1 ;
end proc: # R. J. Mathar, Jun 16 2015
-
(*A060187*)
p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
<< DiscreteMath`Combinatorica`;
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Eulerian[1 + n, m];
t[n_, m_, 2] := f[n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]
A176560
A symmetrical triangle recursion:q=5;t(n,m,0)=Binomial[n,m];t(n,m,1)=Narayana(n,m);t(n,m,2)=Eulerian(n+1,m);t(n,m,q)=t(n,m,g-2)+t(n,m,q-3).
Original entry on oeis.org
1, 1, 1, 1, 6, 1, 1, 16, 16, 1, 1, 35, 85, 35, 1, 1, 71, 351, 351, 71, 1, 1, 140, 1295, 2590, 1295, 140, 1, 1, 274, 4488, 16108, 16108, 4488, 274, 1, 1, 537, 14943, 89409, 157953, 89409, 14943, 537, 1, 1, 1057, 48379, 457711, 1315645, 1315645, 457711, 48379
Offset: 0
{1},
{1, 1},
{1, 6, 1},
{1, 16, 16, 1},
{1, 35, 85, 35, 1},
{1, 71, 351, 351, 71, 1},
{1, 140, 1295, 2590, 1295, 140, 1},
{1, 274, 4488, 16108, 16108, 4488, 274, 1},
{1, 537, 14943, 89409, 157953, 89409, 14943, 537, 1},
{1, 1057, 48379, 457711, 1315645, 1315645, 457711, 48379, 1057, 1},
{1, 2090, 153461, 2208437, 9751973, 15743651, 9751973, 2208437, 153461, 2090, 1}
-
<< DiscreteMath`Combinatorica`
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Binomial[n, m]*Binomial[n + 1, m]/(m + 1);
t[n_, m_, 2] := Eulerian[1 + n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]
A176561
A symmetrical triangle recursion:q=6;t(n,m,0)=Binomial[n,m];t(n,m,1)=Narayana(n,m);t(n,m,2)=Eulerian(n+1,m);t(n,m,q)=t(n,m,g-2)+t(n,m,q-3).
Original entry on oeis.org
1, 1, 1, 1, 7, 1, 1, 18, 18, 1, 1, 38, 90, 38, 1, 1, 75, 360, 360, 75, 1, 1, 145, 1309, 2609, 1309, 145, 1, 1, 280, 4508, 16142, 16142, 4508, 280, 1, 1, 544, 14970, 89464, 158022, 89464, 14970, 544, 1, 1, 1065, 48414, 457794, 1315770, 1315770, 457794, 48414
Offset: 0
{1},
{1, 1},
{1, 7, 1},
{1, 18, 18, 1},
{1, 38, 90, 38, 1},
{1, 75, 360, 360, 75, 1},
{1, 145, 1309, 2609, 1309, 145, 1},
{1, 280, 4508, 16142, 16142, 4508, 280, 1},
{1, 544, 14970, 89464, 158022, 89464, 14970, 544, 1},
{1, 1065, 48414, 457794, 1315770, 1315770, 457794, 48414, 1065, 1},
{1, 2099, 153505, 2208556, 9752182, 15743902, 9752182, 2208556, 153505, 2099, 1}
-
<< DiscreteMath`Combinatorica`
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Binomial[n, m]*Binomial[n + 1, m]/(m + 1);
t[n_, m_, 2] := Eulerian[1 + n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]
Showing 1-3 of 3 results.
Comments