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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":"03e07ebd45428d9afb03367090ac1ab804c0e6992a133e2f1d8e1f09ca2afc67","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-05516f61aeaea779","role":"sequence","name":"Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 5.","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:       1;\n  1:       2,          1;\n  2:      13,         14,           1;\n  3:      84,        249,          86,            1;\n  4:     515,       4166,        3368,          518,            1;\n  5:    3106,      59725,      113240,        40730,         3110,           1;\n  6:   18657,     771886,     3094375,      2412380,       469895,       18662,           1;\n  7:  111968,    9367609,    71838726,    112018175, 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(125*n^2 - 600*n + 725)*x^2 - (125*n^2 - 950*n + 1605)*x^3 - (375*n - 1175)*x^4 - 250*x^5)*D[P[n - 2, x], x] + (35*n - 93 + (50*n^2 - 255*n + 392)*x + (125*n - 325)*x^2 + 50*x^3)*P[n - 1, x] + ((25*n - 70)*x - (25*n - 120)*x^2 - 50*x^3)*D[P[n - 1, x], x])/(5*n - 14 + 10*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(30*n - 54 + (50*n^2 - 255*n + 357)*x + (125*n^3 - 925*n^2 + 2240*n - 1914)*x^2 + (375*n^2 - 1850*n + 2165)*x^3 + (250*n - 450)*x^4)*v[i - 2] - ((25*n - 45)*x + (125*n^2 - 600*n + 725)*x^2 - (125*n^2 - 950*n + 1605)*x^3 - (375*n - 1175)*x^4 - 250*x^5)*deriv(v[i - 2], x) + (35*n - 93 + (50*n^2 - 255*n + 392)*x + (125*n - 325)*x^2 + 50*x^3)*v[i - 1] + ((25*n - 70)*x - (25*n - 120)*x^2 - 50*x^3)*deriv(v[i - 1], x))/(5*n - 14 + 10*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-05516f61aeaea779","steps":["D+1"],"edges":["e-f0b1758f498b5abf"],"text":"A142460 \u2192 D+1"},"headline":"A142460 \u2192 D+1","chain":null}