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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"cb8ec436cf89c459890416e84f657edede3144c34618a18a6c0019a2b72c92be","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-7e652d8a57b6dcff","role":"sequence","name":"Triangle of 4-Eulerian numbers.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-7e652d8a57b6dcff","role":"sequence","name":"Triangle of 4-Eulerian numbers.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"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: 2,    4;\n  1: 2,   17,      16;\n  2: 2,   45,     129,       64;\n  3: 2,  103,     644,      885,       256;\n  4: 2,  221,    2621,     7291,      5641,      1024;\n  5: 2,  459,    9560,    46980,     71130,     34093,       4096;\n  6: 2,  937,   32661,   262310,    678040,    625953,     198113,     16384;\n  7: 2, 1895,  107044,  1335719,   5448170,   8436253,    5109440,   1117541,     65536;\n  8: 2, 3813,  341349,  6388125,  39040521,  93590583,   94226967,  39402951,   6161145,   262144;\n  9: 2, 7651, 1068536, 29225486, 257969936, 908072312, 1400470064, 969896954, 290766854, 33361629, 1048576;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n+2)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (n+4)*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 4*x, P(1,x) = 2 + 17*x + 16*x^2.","mathematica":"P[0, x_] := 2 + 4*x;\nP[1, x_] := 2 + 17*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (n + 2)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (n + 4)*x*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + 4*x; v[2] = 2 + 17*x + 16*x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (n + 2)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (n + 4)*x*v[i - 1] + (x - 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7e652d8a57b6dcff","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-7228e7730f21eb4a"],"text":"A144698 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A144698 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}