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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:    0;\n  1:    2,      2;\n  2:    6,     24,      30,        12;\n  3:   14,    122,     372,       524,        350,         90;\n  4:   30,    480,    2670,      7320,      11130,       9600,        4410,          840;\n  5:   62,   1682,   15180,     67960,     176080,     282372,      285572,       177800,        62370,         9450;\n  6:  126,   5544,   76230,    511812,    2007180,    5021856,     8375724,      9457560,      7161462,      3490200,       990990,       124740;\n  7:  254,  17642,  355572,   3405444,   18948930,   67894974,   165889752,    285144792,    349280514,    303933350,    183883700,     73638180,     17567550,      1891890;\n  8:  510,  54960, 1582590,  20925840,  158632110,  773670240,  2595240270,   6233402160,  10973281770,  14312934960,  13830775530,   9783626160,   4927892970,   1674953280,    344594250,    32432400;\n  9: 1022, 168962, 6828060, 121839920, 1223916260, 7856937864, 34731045804, 110777363760, 262653111360, 471418764860, 646460220692, 678232731792, 540708182300, 322104222440, 138955716900, 41026986000, 7420262850, 620269650;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2+8*x+12*x^2+8*x^3+2*x^4)*P(n-2,x) - (x+4*x^2+6*x^3+4*x^4+x^5)*d/dx P(n-2,x) + (3+8*x+5*x^2)*P(n-1,x) + (x+2*x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(2 + 8*x + 12*x^2 + 8*x^3 + 2*x^4)*P[n - 2, x] - (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*D[P[n - 2, x], x] + (3 + 8*x + 5*x^2)*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] = 0; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 8*x + 12*x^2 + 8*x^3 + 2*x^4)*v[i - 2] - (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*deriv(v[i - 2], x) + (3 + 8*x + 5*x^2)*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"],"edges":["e-f3dd50fc95d394a1"],"text":"A094262 \u2192 D"},"headline":"A094262 \u2192 D","chain":null}