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n>=0.","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,         10,           8;\n  2:        26,        150,         316,          288,            96;\n  3:       252,       2064,        6924,        12184,         11872,          6080,           1280;\n  4:      2568,      27240,      124872,       323256,        517136,        523840,         328320,         116480,          17920;\n  5:     26928,     350688,     2035872,      6940288,      15391152,      23209760,       24112384,       17047296,        7852544,         2128896,          258048;\n  6:    287648,    4440800,    31184128,    131739840,     373001120,     745866592,     1080379840,     1142491392,      875642880,       474485760,       172591104,        37847040,        3784704;\n  7:   3112896,   55575040,   457737536,   2305765248,    7937500736,   19757480704,    36678187200,    51598574464,    55286807552,     44908406784,     27226933248,     11949527040,     3589521408,      660701184,       56229888;\n  8:  34013312,  689376384,  6513848704,  38094173824,  154352456832,  459534900608,  1040018086016,  1825476675456,  2511768453376,   2718687971328,   2307460571136,   1519763976192,   761578954752,   280741773312,    71805566976,   11386552320,     843448320;\n  9: 374416128, 8492659200, 90533993472, 602755424256, 2808892310016, 9730580923392, 25972322718720, 54620986011648, 91762344678144, 124082812639744, 135388294150144, 118930125332480, 83494291234816, 46216232910848, 19728499867648, 6267477032960, 1395532103680, 194367979520, 12745441280;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(8*n^2-24*n+16)*x+(4*n^2-12*n+8)*x^2)*P(n-2,x) + (12*n^2-30*n+12+(28*n^2-70*n+28)*x+(16*n^2-40*n+16)*x^2)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 10*x + 8*x^2, P(2,x) = 26 + 150*x + 316*x^2 + 288*x^3 + 96*x^4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 10*x + 8*x^2;\nP[2, x_] := 26 + 150*x + 316*x^2 + 288*x^3 + 96*x^4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (8*n^2 - 24*n + 16)*x + (4*n^2 - 12*n + 8)*x^2)*P[n - 2, x] + (12*n^2 - 30*n + 12 + (28*n^2 - 70*n + 28)*x + (16*n^2 - 40*n + 16)*x^2)*P[n - 1, x])/(n^2 - 2*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 10*x + 8*x^2; v[3] = 26 + 150*x + 316*x^2 + 288*x^3 + 96*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (8*n^2 - 24*n + 16)*x + (4*n^2 - 12*n + 8)*x^2)*v[i - 2] + (12*n^2 - 30*n + 12 + (28*n^2 - 70*n + 28)*x + (16*n^2 - 40*n + 16)*x^2)*v[i - 1])/(n^2 - 2*n)); 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-97a17a1dc7998ff8","steps":["basis A013609"],"edges":["e-2475c31900bee722"],"text":"A316649 \u2192 basis A013609"},"headline":"A316649 \u2192 basis A013609","chain":null}