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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"7c4e8d13f56ef4ef231f7e65eb4a684bd8390c0f59e64d414458eb77cddbcfb0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-688f70ea6f82c256","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 8).","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-688f70ea6f82c256","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 8).","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:         576,            472,              49,               1;\n  1:        4608,           8128,            1648,              81,               1;\n  2:       36864,         134656,           47680,            4240,             121,              1;\n  3:      294912,        2191360,         1278976,          183360,            9080,            169,             1;\n  4:     2359296,       35356672,        32886784,         7146496,          546560,          17192,           225,           1;\n  5:    18874368,      568066048,       824639488,       261574656,        29008896,        1371776,         29792,         289,          1;\n  6:   150994944,     9107931136,     20359413760,      9195028480,      1421930496,       94854144,       3040128,       48288,        361,        1;\n  7:  1207959552,   145877893120,    497733861376,    314600325120,     66072248320,     5974929408,     265101312,     6130560,      74280,      441,      1;\n  8:  9663676416,  2335254249472,  12091490566144,  10564944265216,   2957490257920,   352868859904,   20820602880,   657457152,   11478720,   109560,    529,   1;\n  9: 77309411328, 37373731667968, 292531027836928, 350169707053056, 128864554582016, 19895195533312, 1518822621184, 62897860608, 1483924992, 20243520, 156112, 625, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 576 + 472*x + 49*x^2 + x^3.","mathematica":"P[0, x_] := 576 + 472*x + 49*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[(8 + x)*P[n - 1, x] + 8*x*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] = 576 + 472*x + 49*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (8 + x)*v[i - 1] + 8*x*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-688f70ea6f82c256","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-2d4f988e0d40946e"],"text":"A075503 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A075503 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}