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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:    3,      4,       1;\n  2:    7,     30,      42,        22,          3;\n  3:   15,    136,     433,       648,        481,        160,          15;\n  4:   31,    510,    2910,      8210,      12960,      11826,        6010,         1470,          105;\n  5:   63,   1744,   16021,     73020,     193070,     317588,      332634,       218596,        84595,        16380,          945;\n  6:  127,   5670,   79002,    537222,    2135133,    5423292,     9212700,     10654092,      8343657,      4285918,      1340010,       214830,        10395;\n  7:  255,  17896,  364393,   3523968,   19800291,   71684760,   177205581,    308843328,    384923613,    342742296,    214277035,     90354880,     23704065,      3243240,       135135;\n  8:  511,  55470, 1610070,  21453370,  163863570,  805396662,  2724185310,   6604150770,  11752457040,  15532188490,  15262069026,  11040969390,   5743195150,   2054021970,    464233770,    55405350,    2027025;\n  9: 1023, 169984, 6912541, 124115940, 1254376240, 8101721116, 36040535448, 115738941732, 276500281830, 500602443900, 693602097178, 737001842764, 597227576616, 363697159540, 161963161360, 50290700460, 9984449475, 1056755700, 34459425;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2+7*x+8*x^2+2*x^3-2*x^4-x^5)*P(n-2,x) - (x+3*x^2+2*x^3-2*x^4-3*x^5-x^6)*d/dx P(n-2,x) + (3+7*x+3*x^2-x^3)*P(n-1,x) + (x+2*x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x + x^2;\nP[n_, x_] := P[n, x] = Expand[-(2 + 7*x + 8*x^2 + 2*x^3 - 2*x^4 - x^5)*P[n - 2, x] - (x + 3*x^2 + 2*x^3 - 2*x^4 - 3*x^5 - x^6)*D[P[n - 2, x], x] + (3 + 7*x + 3*x^2 - x^3)*P[n - 1, x] + (x + 2*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; v[2] = 3 + 4*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 7*x + 8*x^2 + 2*x^3 - 2*x^4 - x^5)*v[i - 2] - (x + 3*x^2 + 2*x^3 - 2*x^4 - 3*x^5 - x^6)*deriv(v[i - 2], x) + (3 + 7*x + 3*x^2 - x^3)*v[i - 1] + (x + 2*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":["D+1"],"edges":["e-b04a126cf24a2c71"],"text":"A094262 \u2192 D+1"},"headline":"A094262 \u2192 D+1","chain":null}