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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":"7529068d9f4fdb3cd0333f21e18f8f7223167d403f054c8e04234f867c5cfb45","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-f3a4ae052af3da90","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 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:         0;\n  1:         5;\n  2:        80,          50;\n  3:       915,        1830,          375;\n  4:      9070,       40260,        27210,         2500;\n  5:     83185,      696820,      1045230,       332740,         15625;\n  6:    727980,    10493310,     29266440,     20986620,       3639900,        93750;\n  7:   6183215,   144545610,    677309565,    903079420,     361364025,     37099290,      546875;\n  8:  51496970,  1874429600,  13818702210,  30704700280,   23031170350,   5623288800,   360478790,    3125000;\n  9: 423303885, 23296568040, 258129652140, 890478630360, 1113098287950, 516259304280, 81537988140, 3386431080, 17578125;\n","recurrence_sentence":"The row polynomials satisfy (3*n-1)*P(n,x) = -(120*n+80+(90*n^2-60*n-80)*x+(27*n^3-27*n^2-18*n+8)*x^2)*P(n-2,x) - ((45*n+30)*x+(27*n^2-36*n-36)*x^2-(27*n^2+9*n-6)*x^3)*d/dx P(n-2,x) + (39*n+2+(18*n^2-12*n+2)*x)*P(n-1,x) + ((9*n-3)*x-(9*n-3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 5;\nP[n_, x_] := P[n, x] = Expand[Together[(-(120*n + 80 + (90*n^2 - 60*n - 80)*x + (27*n^3 - 27*n^2 - 18*n + 8)*x^2)*P[n - 2, x] - ((45*n + 30)*x + (27*n^2 - 36*n - 36)*x^2 - (27*n^2 + 9*n - 6)*x^3)*D[P[n - 2, x], x] + (39*n + 2 + (18*n^2 - 12*n + 2)*x)*P[n - 1, x] + ((9*n - 3)*x - (9*n - 3)*x^2)*D[P[n - 1, x], x])/(3*n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 5; for(n = 2, N, my(i = n + 1); v[i] = (-(120*n + 80 + (90*n^2 - 60*n - 80)*x + (27*n^3 - 27*n^2 - 18*n + 8)*x^2)*v[i - 2] - ((45*n + 30)*x + (27*n^2 - 36*n - 36)*x^2 - (27*n^2 + 9*n - 6)*x^3)*deriv(v[i - 2], x) + (39*n + 2 + (18*n^2 - 12*n + 2)*x)*v[i - 1] + ((9*n - 3)*x - (9*n - 3)*x^2)*deriv(v[i - 1], x))/(3*n - 1)); 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-f3a4ae052af3da90","steps":["D"],"edges":["e-43553b3ae33488df"],"text":"A257624 \u2192 D"},"headline":"A257624 \u2192 D","chain":null}