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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: 1,    4,      4;\n  2: 1,   10,     36,      56,       32;\n  3: 1,   20,    146,     528,     1024,      1024,        416;\n  4: 1,   38,    478,    3016,    11000,     24352,      32416,       23936,         7552;\n  5: 1,   72,   1444,   14140,    81376,    298576,     721984,     1150528,      1166080,       682752,        176128;\n  6: 1,  138,   4224,   60280,   502668,   2714832,   10005408,    25773120,     46552320,     57933056,      47408640,      23000064,        5018624;\n  7: 1,  268,  12214,  244552,  2810956,  20946448,  107987904,   399359232,   1078851456,   2138152448,    3080602112,    3144452096,     2157759488,      893702144,      168968192;\n  8: 1,  526,  35250,  966360, 14823900, 146354520, 1003132640,  4990836320,  18507620160,  51888892160,  110486117888,  177828123648,   213045027840,   184310446080,   108896133120,    39348527104,    6563282944;\n  9: 1, 1040, 102008, 3767524, 75436020, 959955360, 8459217320, 54344440480, 262880626240, 977013635200, 2822041111168, 6362832824320, 11173028056064, 15134028214272, 15514954168320, 11646491336704, 6041406586880, 1936045801472, 288909131776;\n","recurrence_sentence":"The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1-(2*n-9)*x-(14*n-33)*x^2-(38*n-63)*x^3-(50*n-66)*x^4-(32*n-36)*x^5-(8*n-8)*x^6)*P(n-1,x) + (x+8*x^2+26*x^3+44*x^4+41*x^5+20*x^6+4*x^7)*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[Together[((1 - (2*n - 9)*x - (14*n - 33)*x^2 - (38*n - 63)*x^3 - (50*n - 66)*x^4 - (32*n - 36)*x^5 - (8*n - 8)*x^6)*P[n - 1, x] + (x + 8*x^2 + 26*x^3 + 44*x^4 + 41*x^5 + 20*x^6 + 4*x^7)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];\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] = ((1 - (2*n - 9)*x - (14*n - 33)*x^2 - (38*n - 63)*x^3 - (50*n - 66)*x^4 - (32*n - 36)*x^5 - (8*n - 8)*x^6)*v[i - 1] + (x + 8*x^2 + 26*x^3 + 44*x^4 + 41*x^5 + 20*x^6 + 4*x^7)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); 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/(x+1)"],"edges":["e-f8ab0a053d6763c2"],"text":"A094262 \u2192 x \u21a6 x/(x+1)"},"headline":"A094262 \u2192 x \u21a6 x/(x+1)","chain":null}