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Join[{1}, With[{f=Times@@Table[Total[n^Range[0, i]], {i, 4}]}, Table[f, {n, 20}]]] (* _or *) LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {1, 120, 9765, 251680, 3043425, 22661496, 121226245, 510902400, 1799118945, 5507702200, 15072415941}, 30] (* _Harvey P. Dale_, Sep 04 2017 *)
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Join[{1}, With[{f=Times@@Table[Total[n^Range[0, i]], {i, 4}]}, Table[f, {n, 20}]]] (* Harvey P. Dale, Sep 04 2017 *)
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<a href="/index/Rec">Index to sequences with entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
<a href="/index/Rea#recLCCRec">Index to sequences with linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).