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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"79edaec673d77e4a21932db71f76d5f819464d6b9ea9ad2be84e0d0c018c5324","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-65034f179833a7f6","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 10).","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-65034f179833a7f6","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 10).","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:          1100,             730,               61,                1;\n  1:         11000,           15700,             2560,              101,               1;\n  2:        110000,          325000,            92500,             6600,             151,              1;\n  3:       1100000,         6610000,          3100000,           356500,           14150,            211,             1;\n  4:      11000000,       133300000,         99610000,         17360000,         1064000,          26810,           281,            1;\n  5:     110000000,      2677000000,       3121600000,        794010000,        70560000,        2672600,         46480,          361,          1;\n  6:    1100000000,     53650000000,      96325000000,      34882000000,      4322010000,      230916000,       5926200,        75360,        451,        1;\n  7:   11000000000,   1074100000000,    2943400000000,    1491605000000,    250982500000,    18176970000,     645750000,     11955000,     115950,      551,      1;\n  8:  110000000000,  21493000000000,   89376100000000,   62607600000000,  14040730000000,  1341600700000,   63379470000,   1602150000,   22390500,   171050,    661,   1;\n  9: 1100000000000, 429970000000000, 2702776000000000, 2593680100000000, 764644100000000, 94536772000000, 5778163600000, 191551470000, 3617295000, 39495500, 243760, 781, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (10+x)*P(n-1,x) + 10*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1100 + 730*x + 61*x^2 + x^3.","mathematica":"P[0, x_] := 1100 + 730*x + 61*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[(10 + x)*P[n - 1, x] + 10*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] = 1100 + 730*x + 61*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (10 + x)*v[i - 1] + 10*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-65034f179833a7f6","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-f660dc27e1b4f4f9"],"text":"A075505 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A075505 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}