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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":"741a537873124f7973ce285e7df65ef1c9a6c0ff56aa49f6bf392cc8b0d69244","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e6d483569f3966cd","role":"sequence","name":"basis_map_A013610 of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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:      10,         6;\n  2:      72,       126,        36;\n  3:     440,      1368,      1152,        216;\n  4:    2492,     11280,     16884,       9288,       1296;\n  5:   13464,     80424,    176184,     171720,      69984,       7776;\n  6:   70480,    524088,   1521792,    2172312,    1562976,     505440,      46656;\n  7:  360640,   3212736,  11654208,   22087296,   23255424,   13234752,    3545856,    279936;\n  8: 1813968,  18842112,  82113696,  194757696,  272342736,  226514880,  106515648,  24354432,   1679616;\n  9: 9002528, 106863840, 544245696, 1553297472, 2720348064, 3004156512, 2061728640, 825531264, 164602368, 10077696;\n","recurrence_sentence":"The row polynomials satisfy (18*n^2-6*n+(27*n^3+9*n^2-6*n)*x+(54*n^2-72*n)*x^2+(54*n^2-36*n)*x^3+54*n*x^4)*P(n,x) = -(54*n^3-180*n^2-54*n+60+(108*n^3-558*n^2-330*n+408)*x-(162*n^2+630*n-792)*x^2+(216*n^2-468*n+180)*x^3-(108*n+540)*x^4-324*x^5)*P(n-1,x) + (36*n^2+12*n-8+(216*n^2+144*n-48)*x+(378*n^2+630*n+12)*x^2+(162*n^2+1242*n+612)*x^3+(1080*n+1440)*x^4+(324*n+1188)*x^5+324*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(54*n^3 - 180*n^2 - 54*n + 60 + (108*n^3 - 558*n^2 - 330*n + 408)*x - (162*n^2 + 630*n - 792)*x^2 + (216*n^2 - 468*n + 180)*x^3 - (108*n + 540)*x^4 - 324*x^5)*P[n - 1, x] + (36*n^2 + 12*n - 8 + (216*n^2 + 144*n - 48)*x + (378*n^2 + 630*n + 12)*x^2 + (162*n^2 + 1242*n + 612)*x^3 + (1080*n + 1440)*x^4 + (324*n + 1188)*x^5 + 324*x^6)*D[P[n - 1, x], x])/(18*n^2 - 6*n + (27*n^3 + 9*n^2 - 6*n)*x + (54*n^2 - 72*n)*x^2 + (54*n^2 - 36*n)*x^3 + 54*n*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(54*n^3 - 180*n^2 - 54*n + 60 + (108*n^3 - 558*n^2 - 330*n + 408)*x - (162*n^2 + 630*n - 792)*x^2 + (216*n^2 - 468*n + 180)*x^3 - (108*n + 540)*x^4 - 324*x^5)*v[i - 1] + (36*n^2 + 12*n - 8 + (216*n^2 + 144*n - 48)*x + (378*n^2 + 630*n + 12)*x^2 + (162*n^2 + 1242*n + 612)*x^3 + (1080*n + 1440)*x^4 + (324*n + 1188)*x^5 + 324*x^6)*deriv(v[i - 1], x))/(18*n^2 - 6*n + (27*n^3 + 9*n^2 - 6*n)*x + (54*n^2 - 72*n)*x^2 + (54*n^2 - 36*n)*x^3 + 54*n*x^4)); 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-1bffaa6198344f93","steps":["basis A013610","D+1"],"edges":["e-04dfb33ef5bb2a04","e-bca5bdd11d80c941"],"text":"A059473 \u2192 basis A013610 \u2192 D+1"},"headline":"A059473 \u2192 basis A013610 \u2192 D+1","chain":null}