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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"46c44f75e759f104819f12be776d474c440a643ac905499007045b397ff1c6b7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-423e26f84d00b286","role":"sequence","name":"Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-423e26f84d00b286","role":"sequence","name":"Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 2,    2;\n   2: 2,   10,      4;\n   3: 2,   30,     48,        8;\n   4: 2,   74,    308,      216,        16;\n   5: 2,  166,   1472,     2624,       928,        32;\n   6: 2,  354,   5972,    21744,     19968,      3872,        64;\n   7: 2,  734,  21936,   144280,    268960,    141216,     15872,       128;\n   8: 2, 1498,  75604,   830824,   2748944,   2960864,    950720,     64384,      256;\n   9: 2, 3030, 249792,  4347408,  23464704,  44754816,  30067200,   6187776,   259584,     512;\n  10: 2, 6098, 801908, 21269440, 176775488, 545107264, 652371584, 288169984, 39334400, 1042944, 1024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(2*n-4)*x)*P(n-2,x) + (x-2*x^2)*d/dx P(n-2,x) + 2*n*x*P(n-1,x) + (x-2*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + 2*x, P(2,x) = 2 + 10*x + 4*x^2.","mathematica":"P[1, x_] := 2 + 2*x;\nP[2, x_] := 2 + 10*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (2*n - 4)*x)*P[n - 2, x] + (x - 2*x^2)*D[P[n - 2, x], x] + 2*n*x*P[n - 1, x] + (x - 2*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2 + 2*x; v[2] = 2 + 10*x + 4*x^2; for(n = 3, N, my(i = n); v[i] = (1 + (2*n - 4)*x)*v[i - 2] + (x - 2*x^2)*deriv(v[i - 2], x) + 2*n*x*v[i - 1] + (x - 2*x^2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-423e26f84d00b286","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-111ed7b99403026c"],"text":"A142075 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A142075 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}