A165883
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1-x)^(2*n+3)*( Sum_{j >= 0} (2*j+ 1)^n*x^j )*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.
Original entry on oeis.org
1, 1, 2, 1, 1, 10, 26, 10, 1, 1, 34, 287, 508, 287, 34, 1, 1, 102, 2272, 11098, 19134, 11098, 2272, 102, 1, 1, 294, 15493, 169432, 675706, 1042948, 675706, 169432, 15493, 294, 1, 1, 842, 98374, 2151026, 17138559, 55643460, 82178676, 55643460, 17138559, 2151026, 98374, 842, 1
Offset: 0
Irregular triangle begins as:
1;
1, 2, 1;
1, 10, 26, 10, 1;
1, 34, 287, 508, 287, 34, 1;
1, 102, 2272, 11098, 19134, 11098, 2272, 102, 1;
1, 294, 15493, 169432, 675706, 1042948, 675706, 169432, 15493, 294, 1;
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p[n_, x_]:= p[n, x]= (1/x)*(1-x)^(2*n+3)*Sum[(2*k+1)^n*x^k, {k,0,Infinity}]*Sum[k^(n+1)*x^k, {k,0,Infinity}];
Table[CoefficientList[p[n, x], x], {n,0,10}]//Flatten (* modified by G. C. Greubel, Mar 08 2022 *)
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def p(n,x): return (1/x)*(1-x)^(2*n+3)*sum( (2*j+1)^n*x^j for j in (0..2*n+3) )*sum( j^(n+1)*x^j for j in (0..2*n+3) )
def T(n,k): return ( p(n,x) ).series(x, 2*n+1).list()[k]
flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 08 2022
A165890
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = ( (1-x)^(n+1) * Sum_{k >= 0} (2*k+1)^(n-1)*x^k )^2, read by rows.
Original entry on oeis.org
1, 1, -2, 1, 1, 0, -2, 0, 1, 1, 10, 15, -52, 15, 10, 1, 1, 44, 484, -44, -970, -44, 484, 44, 1, 1, 150, 5933, 22792, 466, -58684, 466, 22792, 5933, 150, 1, 1, 472, 58586, 682040, 2085135, -682512, -4287444, -682512, 2085135, 682040, 58586, 472, 1
Offset: 0
Irregular triangle begins as:
1;
1, -2, 1;
1, 0, -2, 0, 1;
1, 10, 15, -52, 15, 10, 1;
1, 44, 484, -44, -970, -44, 484, 44, 1;
1, 150, 5933, 22792, 466, -58684, 466, 22792, 5933, 150, 1;
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p[n_, x_]:= p[n, x]= If[n==0, 1, (2^(n-1)*(1-x)^(n+1)*LerchPhi[x, -n+1, 1/2])^2];
Table[CoefficientList[p[n, x], x], {n,0,12}]//Flatten (* modified by G. C. Greubel, Mar 09 2022 *)
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def p(n,x): return (1-x)^(2*n+2)*sum( (2*j+1)^(n-1)*x^j for j in (0..2*n+2) )^2
def T(n,k): return ( p(n,x) ).series(x, 2*n+2).list()[k]
flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2022
A165891
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.
Original entry on oeis.org
1, 1, 2, 1, 1, 6, 10, 6, 1, 1, 14, 47, 68, 47, 14, 1, 1, 30, 176, 450, 606, 450, 176, 30, 1, 1, 62, 597, 2392, 5162, 6612, 5162, 2392, 597, 62, 1, 1, 126, 1926, 11382, 35967, 69132, 85492, 69132, 35967, 11382, 1926, 126, 1, 1, 254, 6043, 50892, 223785, 600546, 1060411, 1277096, 1060411, 600546, 223785, 50892, 6043, 254, 1
Offset: 0
Irregular triangle begins as:
1;
1, 2, 1;
1, 6, 10, 6, 1;
1, 14, 47, 68, 47, 14, 1;
1, 30, 176, 450, 606, 450, 176, 30, 1;
1, 62, 597, 2392, 5162, 6612, 5162, 2392, 597, 62, 1;
1, 126, 1926, 11382, 35967, 69132, 85492, 69132, 35967, 11382, 1926, 126, 1;
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p[n_, x_]:= p[n, x]= (1/x)*(1+x)^n*(1-x)^(n+2)*PolyLog[-n-1, x];
Table[CoefficientList[p[n, x], x], {n,0,12}]//Flatten (* modified by G. C. Greubel, Mar 09 2022 *)
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def p(n,x): return (1/x)*(1+x)^n*(1-x)^(n+2)*polylog(-n-1, x)
def T(n,k): return ( p(n,x) ).series(x, 2*n+1).list()[k]
flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2022
A382232
Irregular triangle read by rows: T(n,k) = [x^k] (1+x) * A_n(x)^2, where A_n(x) is the n-th Eulerian polynomial.
Original entry on oeis.org
1, 1, 1, 1, 1, 3, 3, 1, 1, 9, 26, 26, 9, 1, 1, 23, 165, 387, 387, 165, 23, 1, 1, 53, 860, 4292, 9194, 9194, 4292, 860, 53, 1, 1, 115, 3967, 38885, 160778, 314654, 314654, 160778, 38885, 3967, 115, 1, 1, 241, 17022, 307454, 2291375, 8041695, 14743812, 14743812, 8041695, 2291375, 307454, 17022, 241, 1
Offset: 0
Irregular triangle begins:
1, 1;
1, 1;
1, 3, 3, 1;
1, 9, 26, 26, 9, 1;
1, 23, 165, 387, 387, 165, 23, 1;
1, 53, 860, 4292, 9194, 9194, 4292, 860, 53, 1;
...
- Ryuichi Sakamoto, The h*-polynomial of the cut polytope of K_{2,m} in the lattice spanned by its vertices, arXiv:1904.10667 [math.CO], 2019.
- Ryuichi Sakamoto, The h*-polynomial of the cut polytope of K_{2,m} in the lattice spanned by its vertices, Journal of Integer Sequences, Vol. 23, 2020, #20.7.5.
- OEIS Wiki, Eulerian polynomials.
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a(n) = sum(k=0, n, k!*stirling(n, k, 2)*(x-1)^(n-k));
T(n, k) = polcoef((1+x)*a(n)^2, k);
for(n=0, 7, for(k=0, 2*(n+0^n)-1, print1(T(n, k), ", ")));
Showing 1-4 of 4 results.