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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":"8f363a5458ba44ee50359dc20384b84d9ce1ab56a950351620eea4598dc26f6c","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-63490e7c6342a0ed","role":"sequence","name":"veronese_section_2_0 of Triangle of coefficients in expansion of (1+3*x)^n.","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:   1;\n  2:  10,     9;\n  3:  28,    27;\n  4:  55,   216,     81;\n  5:  91,   900,    405;\n  6: 136,  2565,   3402,    729;\n  7: 190,  5859,  18144,   5103;\n  8: 253, 11592,  66906,  46656,  6561;\n  9: 325, 20736, 193914, 297432, 59049;\n","recurrence_sentence":"The row polynomials satisfy (9*n^3-36*n^2+47*n-20+(36*n^2-130*n+94)*x+(36*n-36)*x^2)*P(n,x) = (9*n^3-18*n^2+11*n-2+(81*n^3-288*n^2+95*n+112)*x+(324*n^2-810*n+486)*x^2+(324*n-324)*x^3)*P(n-1,x) + ((18*n^2-18*n-4)*x-(162*n^2-234*n+8)*x^2-(648*n-468)*x^3-648*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 18*n^2 + 11*n - 2 + (81*n^3 - 288*n^2 + 95*n + 112)*x + (324*n^2 - 810*n + 486)*x^2 + (324*n - 324)*x^3)*P[n - 1, x] + ((18*n^2 - 18*n - 4)*x - (162*n^2 - 234*n + 8)*x^2 - (648*n - 468)*x^3 - 648*x^4)*D[P[n - 1, x], x])/(9*n^3 - 36*n^2 + 47*n - 20 + (36*n^2 - 130*n + 94)*x + (36*n - 36)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^3 - 18*n^2 + 11*n - 2 + (81*n^3 - 288*n^2 + 95*n + 112)*x + (324*n^2 - 810*n + 486)*x^2 + (324*n - 324)*x^3)*v[i - 1] + ((18*n^2 - 18*n - 4)*x - (162*n^2 - 234*n + 8)*x^2 - (648*n - 468)*x^3 - 648*x^4)*deriv(v[i - 1], x))/(9*n^3 - 36*n^2 + 47*n - 20 + (36*n^2 - 130*n + 94)*x + (36*n - 36)*x^2)); 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-6a5af5e1168bb679","steps":["Veronese 2","D+1"],"edges":["e-a3d90d8bb42bdd7e","e-709cb4eedb5a58dc"],"text":"A013610 \u2192 Veronese 2 \u2192 D+1"},"headline":"A013610 \u2192 Veronese 2 \u2192 D+1","chain":null}