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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"d2345dc9d08479640c3e757e93610f5ad48bfa3c42130f8628390d44063cfe13","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e48eea24800097e1","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(n-2, k-1) + n*binomial(n-1, k-1), 1 <= k <= n, starting with T(1, 1) = 1.","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:    5,     4;\n  2:   14,    26,     12;\n  3:   36,   104,    100,     32;\n  4:   88,   344,    504,    328,     80;\n  5:  208,  1024,   2016,   1984,    976,    192;\n  6:  480,  2848,   7040,   9280,   6880,   2720,    448;\n  7: 1088,  7552,  22464,  37120,  36800,  21888,   7232,   1024;\n  8: 2432, 19328,  67200, 133504, 165760, 131712,  65408,  18560,  2304;\n  9: 5376, 48128, 191488, 444416, 663040, 659456, 437248, 186368, 46336, 5120;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(8*n-16)*x+(4*n-8)*x^2)*P(n-2,x) + (4*n-8+(4*n-8)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 4*x, P(2,x) = 14 + 26*x + 12*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 4*x;\nP[2, x_] := 14 + 26*x + 12*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (8*n - 16)*x + (4*n - 8)*x^2)*P[n - 2, x] + (4*n - 8 + (4*n - 8)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 4*x; v[3] = 14 + 26*x + 12*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (8*n - 16)*x + (4*n - 8)*x^2)*v[i - 2] + (4*n - 8 + (4*n - 8)*x)*v[i - 1])/(n - 2)); 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-e48eea24800097e1","steps":["basis A013609"],"edges":["e-167670445364346d"],"text":"A123097 \u2192 basis A013609"},"headline":"A123097 \u2192 basis A013609","chain":null}