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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"097b1c7a5f437f9d69ef4e0f80545e7917f5ea2bb538174d650dd58eef8ab0a0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0026e6b972fa04d4","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 9).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"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:           108,             29,               1;\n  1:          1296,            675,              57,              1;\n  2:         17496,          14985,            2295,             94,              1;\n  3:        262440,         334611,           81405,           5805,            140,            1;\n  4:       4251528,        7669809,         2740311,         300510,          12285,          195,           1;\n  5:      72275976,      181575675,        90522117,       14222061,         873180,        23058,         259,          1;\n  6:    1262703816,     4434875145,      2984041215,      641232774,       55250181,      2153466,       39690,        332,        1;\n  7:   22384294920,   111247075971,     98999486685,    28169030205,     3259898460,    175487067,     4711770,      63990,      414,      1;\n  8:  399817944648,  2850080350689,   3318227790471,  1221607483470,   184115077245,  13117440105,   480442347,    9408960,    98010,    505,   1;\n  9: 7168828728456, 74181338131275, 112491284045877, 52713622692621, 10114842151260, 925725580956, 44350739433, 1173283947, 17481420, 144045, 605, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(162-63*x-8*x^2)*P(n-2,x) - (81*x-72*x^2)*d/dx P(n-2,x) + (27-7*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 108 + 29*x + x^2, P(1,x) = 1296 + 675*x + 57*x^2 + x^3.","mathematica":"P[0, x_] := 108 + 29*x + x^2;\nP[1, x_] := 1296 + 675*x + 57*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[-(162 - 63*x - 8*x^2)*P[n - 2, x] - (81*x - 72*x^2)*D[P[n - 2, x], x] + (27 - 7*x)*P[n - 1, x] + 9*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] = 108 + 29*x + x^2; v[2] = 1296 + 675*x + 57*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(162 - 63*x - 8*x^2)*v[i - 2] - (81*x - 72*x^2)*deriv(v[i - 2], x) + (27 - 7*x)*v[i - 1] + 9*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-0026e6b972fa04d4","steps":["D+1"],"edges":["e-5232b598e65b8f46"],"text":"A075504 \u2192 D+1"},"headline":"A075504 \u2192 D+1","chain":null}