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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"faa45b2850dedfd7126f9e27b86995b8daa91728b56e98ab5c9dafe8f3c7b0f6","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-e7d0ffa461d9dbc4","role":"sequence","name":"Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).","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,      8;\n  2:    18,     52,     36;\n  3:    56,    228,    300,     128;\n  4:   160,    832,   1584,    1312,     400;\n  5:   432,   2720,   6720,    8160,    4880,    1152;\n  6:  1120,   8256,  24960,   39680,   35040,   16320,    3136;\n  7:  2816,  23744,  84672,  165760,  192640,  133056,   50624,    8192;\n  8:  6912,  65536, 268800,  623616,  896000,  817152,  462336,  148480,  20736;\n  9: 16640, 175104, 811008, 2171904, 3709440, 4193280, 3139584, 1502208, 417024, 51200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+24*x+24*x^2+8*x^3)*P(n-3,x) - (12+24*x+12*x^2)*P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 8*x, P(2,x) = 18 + 52*x + 36*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 8*x;\nP[2, x_] := 18 + 52*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[(8 + 24*x + 24*x^2 + 8*x^3)*P[n - 3, x] - (12 + 24*x + 12*x^2)*P[n - 2, x] + (6 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 8*x; v[3] = 18 + 52*x + 36*x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 24*x + 24*x^2 + 8*x^3)*v[i - 3] - (12 + 24*x + 12*x^2)*v[i - 2] + (6 + 6*x)*v[i - 1]); 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-e7d0ffa461d9dbc4","steps":["basis A013609"],"edges":["e-81229512ca454bf5"],"text":"A125092 \u2192 basis A013609"},"headline":"A125092 \u2192 basis A013609","chain":null}