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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":"45a1711e52edf9be76878604a08a811b4500ccab9b8ca04c474320cbd4be09ab","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-fabb0a68e001a863","role":"sequence","name":"Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows.","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:   9,    10,       1;\n  3:  28,    81,      30,       1;\n  4:  65,   496,     408,      68,        1;\n  5: 126,  2125,    4000,    1500,      130,       1;\n  6: 217,  6966,   27375,   21500,     4455,     222,      1;\n  7: 344, 18865,  137886,  214375,    89180,   11319,    350,     1;\n  8: 513, 44416,  548800, 1547616,  1221080,  307328,  25536,   520,   1;\n  9: 730, 94041, 1824768, 8594208, 12002256, 5556600, 919296, 52488, 738, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^5-3*n^4+3*n^3+(3*n^4-9*n^3+4*n^2)*x+(3*n^3-5*n^2)*x^2+n^2*x^3)*P(n,x) = (n^5+n^2+(10*n^5-42*n^4+51*n^3-28*n^2+6*n)*x+(30*n^4-120*n^3+147*n^2-75*n+12)*x^2+(30*n^3-84*n^2+78*n-24)*x^3+(10*n^2-15*n+6)*x^4)*P(n-1,x) + ((3*n^4+9*n^3-15*n^2)*x-(15*n^4-60*n^3+22*n^2+42*n)*x^2-(45*n^3-152*n^2+105*n+12)*x^3-(45*n^2-102*n+48)*x^4-(15*n-18)*x^5)*d/dx P(n-1,x) + ((6*n^3-8*n^2)*x^2+(6*n^3+10*n^2-24*n)*x^3+(18*n^2-6*n-12)*x^4+(18*n-6)*x^5+6*x^6)*d^2/dx^2 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[((n^5 + n^2 + (10*n^5 - 42*n^4 + 51*n^3 - 28*n^2 + 6*n)*x + (30*n^4 - 120*n^3 + 147*n^2 - 75*n + 12)*x^2 + (30*n^3 - 84*n^2 + 78*n - 24)*x^3 + (10*n^2 - 15*n + 6)*x^4)*P[n - 1, x] + ((3*n^4 + 9*n^3 - 15*n^2)*x - (15*n^4 - 60*n^3 + 22*n^2 + 42*n)*x^2 - (45*n^3 - 152*n^2 + 105*n + 12)*x^3 - (45*n^2 - 102*n + 48)*x^4 - (15*n - 18)*x^5)*D[P[n - 1, x], x] + ((6*n^3 - 8*n^2)*x^2 + (6*n^3 + 10*n^2 - 24*n)*x^3 + (18*n^2 - 6*n - 12)*x^4 + (18*n - 6)*x^5 + 6*x^6)*D[P[n - 1, x], {x, 2}])/(n^5 - 3*n^4 + 3*n^3 + (3*n^4 - 9*n^3 + 4*n^2)*x + (3*n^3 - 5*n^2)*x^2 + n^2*x^3)]];\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] = ((n^5 + n^2 + (10*n^5 - 42*n^4 + 51*n^3 - 28*n^2 + 6*n)*x + (30*n^4 - 120*n^3 + 147*n^2 - 75*n + 12)*x^2 + (30*n^3 - 84*n^2 + 78*n - 24)*x^3 + (10*n^2 - 15*n + 6)*x^4)*v[i - 1] + ((3*n^4 + 9*n^3 - 15*n^2)*x - (15*n^4 - 60*n^3 + 22*n^2 + 42*n)*x^2 - (45*n^3 - 152*n^2 + 105*n + 12)*x^3 - (45*n^2 - 102*n + 48)*x^4 - (15*n - 18)*x^5)*deriv(v[i - 1], x) + ((6*n^3 - 8*n^2)*x^2 + (6*n^3 + 10*n^2 - 24*n)*x^3 + (18*n^2 - 6*n - 12)*x^4 + (18*n - 6)*x^5 + 6*x^6)*deriv(deriv(v[i - 1], x), x))/(n^5 - 3*n^4 + 3*n^3 + (3*n^4 - 9*n^3 + 4*n^2)*x + (3*n^3 - 5*n^2)*x^2 + n^2*x^3)); 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-fabb0a68e001a863","steps":["D+1"],"edges":["e-281e931411b5d5d5"],"text":"A181543 \u2192 D+1"},"headline":"A181543 \u2192 D+1","chain":null}