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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"51764dcd1183a30137e7507dc0b83e1674c8c968df3ef17752634dee1c6a07c5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-4c402d5a817396fa","role":"sequence","name":"Triangle of 3-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-4c402d5a817396fa","role":"sequence","name":"Triangle of 3-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,    3;\n  1: 2,   13,      9;\n  2: 2,   35,     76,       27;\n  3: 2,   81,    393,      403,        81;\n  4: 2,  175,   1640,     3510,      1990,       243;\n  5: 2,  365,   6089,    23554,     27124,      9337,       729;\n  6: 2,  747,  21068,   135525,    273810,    189733,     42208,      2187;\n  7: 2, 1513,  69681,   705869,   2297305,   2756043,   1235027,    185599,     6561;\n  8: 2, 3047, 223672,  3434312,  17018372,  32368798,  25000808,   7618448,   799258,   19683;\n  9: 2, 6117, 703521, 15920580, 115409904, 328311054, 399011838, 209758884, 45102150, 3387301, 59049;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n+1)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (n+3)*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 3*x, P(1,x) = 2 + 13*x + 9*x^2.","mathematica":"P[0, x_] := 2 + 3*x;\nP[1, x_] := 2 + 13*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (n + 1)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (n + 3)*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 + 3*x; v[2] = 2 + 13*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (n + 1)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (n + 3)*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-4c402d5a817396fa","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-b4867a0cc7e138f4"],"text":"A144697 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A144697 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}
