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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"a3aea6d1f26b432e8117abc368db80f21bf2529c82e488542da024efbd52678f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-7b6a59413dc1b477","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.","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-7b6a59413dc1b477","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.","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;\n  1: 2,     1;\n  2: 2,     6,      1;\n  3: 2,    23,     16,       1;\n  4: 2,    76,    144,      38,        1;\n  5: 2,   237,   1006,     683,       84,       1;\n  6: 2,   722,   6157,    8676,     2768,     178,       1;\n  7: 2,  2179,  34938,   90640,    58970,   10179,     368,      1;\n  8: 2,  6552, 189400,  838828,   975900,  345164,   35144,    750,    1;\n  9: 2, 19673, 997780, 7167938, 13733458, 8620019, 1827526, 116315, 1516, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n-2)*x)*P(n-2,x) + (2*x-x^2)*d/dx P(n-2,x) + n*x*P(n-1,x) + (2*x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 2)*x)*P[n - 2, x] + (2*x - x^2)*D[P[n - 2, x], x] + n*x*P[n - 1, x] + (2*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; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + (n - 2)*x)*v[i - 2] + (2*x - x^2)*deriv(v[i - 2], x) + n*x*v[i - 1] + (2*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-7b6a59413dc1b477","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-9fcf9816b8c5606e"],"text":"A102365 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A102365 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}