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(x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:           1;\n  1:           3,            1;\n  2:          18,           16,             3;\n  3:         174,          280,           135,            18;\n  4:        2474,         6000,          5095,          1728,           180;\n  5:       48824,       160980,        202585,        119193,         31500,          2700;\n  6:     1281104,      5385192,       9118576,       7867584,       3574053,        777600,         56700;\n  7:    43223312,    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= 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (2*n^5 - 2*n^4 + 2*n^3 + 2*n^2 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*v[i - 2] + (4*n^3 + 4*n + (4*n^3 - 4*n)*x)*v[i - 1])/(4*n - 4)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ecd77b030b8fab94","steps":["basis A007318","D+1"],"edges":["e-8bd02eafb0f0a53d","e-cb42a64799910377"],"text":"A128813 \u2192 basis A007318 \u2192 D+1"},"headline":"A128813 \u2192 basis A007318 \u2192 D+1","chain":null}