A165889
Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1-x)^(2*n+4)*( Sum_{j >= 0} j^(n+1)*x^j )^2/x^2, read by rows.
Original entry on oeis.org
1, 1, 2, 1, 1, 8, 18, 8, 1, 1, 22, 143, 244, 143, 22, 1, 1, 52, 808, 3484, 5710, 3484, 808, 52, 1, 1, 114, 3853, 35032, 125746, 188908, 125746, 35032, 3853, 114, 1, 1, 240, 16782, 290672, 2000703, 6040992, 8702820, 6040992, 2000703, 290672, 16782, 240, 1
Offset: 0
Irregular triangle begins as:
1;
1, 2, 1;
1, 8, 18, 8, 1;
1, 22, 143, 244, 143, 22, 1;
1, 52, 808, 3484, 5710, 3484, 808, 52, 1;
1, 114, 3853, 35032, 125746, 188908, 125746, 35032, 3853, 114, 1;
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p[n_, x_]:= p[n, x]= (1/x^2)*(1-x)^(2*n+4)*Sum[k^(n+1)*x^k, {k, 0, Infinity}]^2;
Table[CoefficientList[p[n, x], x], {n,0,12}]//Flatten (* modified by G. C. Greubel, Mar 09 2022 *)
-
def p(n,x): return (1/x^2)*(1-x)^(2*n+4)*sum( j^(n+1)*x^j for j in (0..2*n+4) )^2
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)])
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 *)
-
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 *)
-
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
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