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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,          5,           2;\n  2:        26,         75,          79,          36,            6;\n  3:       252,       1032,        1731,        1523,          742,          190,           20;\n  4:      2568,      13620,       31218,       40407,        32321,        16370,         5130,          910,           70;\n  5:     26928,     175344,      508968,      867536,       961947,       725305,       376756,       133182,        30674,         4158,          252;\n  6:    287648,    2220400,     7796032,    16467480,     23312570,     23308331,     16880935,      8925714,      3420480,       926730,       168546,       18480,         924;\n  7:   3112896,   27787520,   114434384,   288220656,    496093796,    617421272,    573096675,    403113863,    215964092,     87711732,     26588802,     5834730,      876348,      80652,       3432;\n  8:  34013312,  344688192,  1628462176,  4761771728,   9647028552,  14360465644,  16250282594,  14261536527,   9811595521,   5309937444,   2253379464,   742072254,   185932362,   34270236,    4382664,    347490,    12870;\n  9: 374416128, 4246329600, 22633498368, 75344428032, 175555769376, 304080653856, 405817542480, 426726453216, 358446658899, 242349243437, 132215131006, 58071350260, 20384348446, 5641629994, 1204132072, 191268220, 21294130, 1482910, 48620;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(4*n^2-12*n+8)*x+(n^2-3*n+2)*x^2)*P(n-2,x) + (12*n^2-30*n+12+(14*n^2-35*n+14)*x+(4*n^2-10*n+4)*x^2)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 5*x + 2*x^2, P(2,x) = 26 + 75*x + 79*x^2 + 36*x^3 + 6*x^4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 5*x + 2*x^2;\nP[2, x_] := 26 + 75*x + 79*x^2 + 36*x^3 + 6*x^4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (4*n^2 - 12*n + 8)*x + (n^2 - 3*n + 2)*x^2)*P[n - 2, x] + (12*n^2 - 30*n + 12 + (14*n^2 - 35*n + 14)*x + (4*n^2 - 10*n + 4)*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 + 5*x + 2*x^2; v[3] = 26 + 75*x + 79*x^2 + 36*x^3 + 6*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (4*n^2 - 12*n + 8)*x + (n^2 - 3*n + 2)*x^2)*v[i - 2] + (12*n^2 - 30*n + 12 + (14*n^2 - 35*n + 14)*x + (4*n^2 - 10*n + 4)*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 A007318"],"edges":["e-a20b01203cd2c334"],"text":"A316649 \u2192 basis A007318"},"headline":"A316649 \u2192 basis A007318","chain":null}