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A165883
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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.
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4
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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
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OFFSET
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0,3
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LINKS
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FORMULA
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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 ).
T(n, k) = [x^k]( p(n, x) ), where p(n, x) = 2^n*(1-x)^(2*n+3)*LerchPhi(x, -n, 1/2)*PolyLog(-n-1, x)/x.
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EXAMPLE
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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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MATHEMATICA
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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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PROG
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(Sage)
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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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