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DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     2,      1;\n   3:     6,      6,      1;\n   4:    18,     30,     11,      1;\n   5:    54,    138,     85,     17,      1;\n   6:   162,    606,    563,    187,     24,      1;\n   7:   486,   2586,   3421,   1685,    355,     32,     1;\n   8:  1458,  10830,  19691,  13531,   4170,    611,    41,    1;\n   9:  4374,  44778, 109285, 100877,  42721,   9058,   980,   51,  1;\n  10: 13122, 183486, 591203, 714547, 399924, 115185, 17878, 1490, 62, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (3*n-6+(n-2)*x)*P(n-1,x) + (n-2)*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n - 6 + (n - 2)*x)*P[n - 1, x] + (n - 2)*x*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = ((3*n - 6 + (n - 2)*x)*v[i - 1] + (n - 2)*x*deriv(v[i - 1], x))/(n - 2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f2a2a3d7ea29bcbf","steps":["basis A008277"],"edges":["e-c4fc16ce48ccfd20"],"text":"A200139 \u2192 basis A008277"},"headline":"A200139 \u2192 basis A008277","chain":null}