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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":"4f2923abf0c72d70fb1a344ece02ae6ba117f8415cfbb6603a888b41040c8b5e","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-7c7a913d07444057","role":"sequence","name":"basis_map_A013609 of Even row Pascal-square read by antidiagonals.","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:     7,      4;\n  2:    28,     44,      12;\n  3:    97,    252,     184,      32;\n  4:   315,   1148,    1404,     640,      80;\n  5:   984,   4632,    8040,    6208,    2000,     192;\n  6:  2993,  17304,   39120,   43296,   23856,    5824,     448;\n  7:  8927,  61268,  171144,  248480,  197920,   83328,   16128,   1024;\n  8: 26228, 208516,  694548, 1252544, 1317040,  807552,  271488,  43008,   2304;\n  9: 76145, 688404, 2665352, 5754560, 7551120, 6146240, 3028992, 838656, 111360, 5120;\n","recurrence_sentence":"The row polynomials satisfy (64*n^3-16*n^2-36*n+(128*n^2-88*n)*x+64*n*x^2)*P(n,x) = (256*n^3+64*n^2-264*n+28+(384*n^3+544*n^2-680*n+80)*x+(768*n^2-144*n-16)*x^2+(384*n-128)*x^3)*P(n-1,x) - (80*n^2+100*n+5+(288*n^2+520*n+68)*x+(256*n^2+896*n+276)*x^2+(512*n+448)*x^3+256*x^4)*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[((256*n^3 + 64*n^2 - 264*n + 28 + (384*n^3 + 544*n^2 - 680*n + 80)*x + (768*n^2 - 144*n - 16)*x^2 + (384*n - 128)*x^3)*P[n - 1, x] - (80*n^2 + 100*n + 5 + (288*n^2 + 520*n + 68)*x + (256*n^2 + 896*n + 276)*x^2 + (512*n + 448)*x^3 + 256*x^4)*D[P[n - 1, x], x])/(64*n^3 - 16*n^2 - 36*n + (128*n^2 - 88*n)*x + 64*n*x^2)]];\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] = ((256*n^3 + 64*n^2 - 264*n + 28 + (384*n^3 + 544*n^2 - 680*n + 80)*x + (768*n^2 - 144*n - 16)*x^2 + (384*n - 128)*x^3)*v[i - 1] - (80*n^2 + 100*n + 5 + (288*n^2 + 520*n + 68)*x + (256*n^2 + 896*n + 276)*x^2 + (512*n + 448)*x^3 + 256*x^4)*deriv(v[i - 1], x))/(64*n^3 - 16*n^2 - 36*n + (128*n^2 - 88*n)*x + 64*n*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-79ebafb266da3b41","steps":["basis A013609","D+1"],"edges":["e-1a56a7670c2d3c41","e-bb69cd4813efcc98"],"text":"A172431 \u2192 basis A013609 \u2192 D+1"},"headline":"A172431 \u2192 basis A013609 \u2192 D+1","chain":null}