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If a node has an empty set of labels then it must have at least two children.","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:            4,             4,              1;\n  2:           32,            72,             60,             22,               3;\n  3:          416,          1472,           2144,           1648,             706,             160,              15;\n  4:         7552,         36480,          76320,          90400,           66360,           30936,            8950,            1470,             105;\n  5:       176128,       1078528,        2947072,        4734400,         4954240,         3530224,         1735504,          581476,          127120,           16380,            945;\n  6:      5018624,      37223424,      125637120,      255247104,       347751936,       334821312,       233675232,       119144880,        44059932,        11527768,        2026080,         214830,          10395;\n  7:    168968192,    1471852544,     5915737088,    14544316416,     24442102272,     29707378176,     26935746432,     18512496384,      9693451968,      3848949616,     1141043260,      244949320,       36001350,       3243240,       135135;\n  8:   6563282944,   65664000000,   306262179840,   884078520320,   1768152468480,   2598370274304,   2902733191680,   2514999893760,   1708254504000,    912773758880,   382465002816,   124369234440,    30772269220,    5601315720,    707476770,     55405350,     2027025;\n  9: 288909131776, 3264318570496, 17331725123584, 57463636582400, 133337172776960, 229965861994496, 305555623289856, 319847287916544, 267425289790080, 179952603113600, 97699511273024, 42699886231904, 14913395202376, 4106416840240, 871921910860, 137771694060, 15256741500, 1056755700, 34459425;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+4*x+x^2)*P(n-1,x) + (4+8*x+5*x^2+x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4 + 4*x + x^2)*P[n - 1, x] + (4 + 8*x + 5*x^2 + x^3)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4 + 4*x + x^2)*v[i - 1] + (4 + 8*x + 5*x^2 + x^3)*deriv(v[i - 1], x)); 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-f712c326ca8e5415","steps":["x \u21a6 x+1"],"edges":["e-2263ada65e6ad839"],"text":"A094262 \u2192 x \u21a6 x+1"},"headline":"A094262 \u2192 x \u21a6 x+1","chain":null}