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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":"57cd6564616e3dc9941bda1fb97747fc5da959d94030f972e5f229404e5510ac","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-78b19c28b9f05152","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=0 and 0<=k<=2n.","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: 12, 13, 1;\n 3: 31, 90, 33, 1;\n 4: 59, 448, 369, 62, 1;\n 5: 96, 1465, 2740, 1065, 100, 1;\n 6: 142, 3681, 13737, 11069, 2475, 147, 1;\n 7: 197, 7798, 50400, 77665, 34538, 4977, 203, 1;\n 8: 261, 14680, 147994, 392988, 321155, 90188, 9030, 268, 1;\n 9: 334, 25353, 369972, 1544286, 2137698, 1071207, 206724, 15174, 342, 1;\n","recurrence_sentence":"The row polynomials satisfy (9*n^3-34*n^2+43*n-18-(13*n^3-66*n^2+137*n-84)*x+(4*n^3-50*n^2+144*n-98)*x^2+(8*n^2-54*n+46)*x^3+(4*n-4)*x^4)*P(n,x) = (9*n^3-16*n^2+9*n-2+(59*n^3-224*n^2+77*n+88)*x-(55*n^3-350*n^2+553*n-258)*x^2+(12*n^3-174*n^2+506*n-344)*x^3+(24*n^2-166*n+142)*x^4+(12*n-12)*x^5)*P(n-1,x) + ((18*n^2-18*n-4)*x-(134*n^2-190*n+8)*x^2+(74*n^2-482*n+316)*x^3-(8*n^2-176*n+520)*x^4-(16*n-124)*x^5-8*x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 16*n^2 + 9*n - 2 + (59*n^3 - 224*n^2 + 77*n + 88)*x - (55*n^3 - 350*n^2 + 553*n - 258)*x^2 + (12*n^3 - 174*n^2 + 506*n - 344)*x^3 + (24*n^2 - 166*n + 142)*x^4 + (12*n - 12)*x^5)*P[n - 1, x] + ((18*n^2 - 18*n - 4)*x - (134*n^2 - 190*n + 8)*x^2 + (74*n^2 - 482*n + 316)*x^3 - (8*n^2 - 176*n + 520)*x^4 - (16*n - 124)*x^5 - 8*x^6)*D[P[n - 1, x], x])/(9*n^3 - 34*n^2 + 43*n - 18 - (13*n^3 - 66*n^2 + 137*n - 84)*x + (4*n^3 - 50*n^2 + 144*n - 98)*x^2 + (8*n^2 - 54*n + 46)*x^3 + (4*n - 4)*x^4)]];\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] = ((9*n^3 - 16*n^2 + 9*n - 2 + (59*n^3 - 224*n^2 + 77*n + 88)*x - (55*n^3 - 350*n^2 + 553*n - 258)*x^2 + (12*n^3 - 174*n^2 + 506*n - 344)*x^3 + (24*n^2 - 166*n + 142)*x^4 + (12*n - 12)*x^5)*v[i - 1] + ((18*n^2 - 18*n - 4)*x - (134*n^2 - 190*n + 8)*x^2 + (74*n^2 - 482*n + 316)*x^3 - (8*n^2 - 176*n + 520)*x^4 - (16*n - 124)*x^5 - 8*x^6)*deriv(v[i - 1], x))/(9*n^3 - 34*n^2 + 43*n - 18 - (13*n^3 - 66*n^2 + 137*n - 84)*x + (4*n^3 - 50*n^2 + 144*n - 98)*x^2 + (8*n^2 - 54*n + 46)*x^3 + (4*n - 4)*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-b30e7a5b4198051a","steps":["Veronese 2","D+1"],"edges":["e-864f7f13d69989fc","e-090b2bbd9f1b77c0"],"text":"A272866 \u2192 Veronese 2 \u2192 D+1"},"headline":"A272866 \u2192 Veronese 2 \u2192 D+1","chain":null}