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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":"0f64968a987f4d9230dbe28ffd9bcb18a9fe6bdfa8e351d93ec0df5bcd2dc9a8","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-da6c9018f05787b2","role":"sequence","name":"basis_map_A013609 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:      0;\n  1:      2;\n  2:     14,       8;\n  3:     70,      88,      24;\n  4:    310,     608,     360,      64;\n  5:   1284,    3432,    3192,    1216,     160;\n  6:   5100,   17280,   22056,   13184,    3680,     384;\n  7:  19682,   80808,  131496,  108096,   47200,   10368,    896;\n  8:  74378,  358832,  710808,  746944,  448480,  153600,  27776,  2048;\n  9: 276610, 1533544, 3582840, 4595072, 3530240, 1660800, 466816, 71680, 4608;\n","recurrence_sentence":"The row polynomials satisfy -(6*n^2-28*n+22-(12*n^2-4*n-8)*x+(8*n-8)*x^2)*P(n,x) = -(48*n^2-158*n+54+(48*n^2-304*n+152)*x-(104*n-136)*x^2+32*x^3)*P(n-1,x) + (39*n-26+(138*n-118)*x+(132*n-180)*x^2+(24*n-104)*x^3-16*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(48*n^2 - 158*n + 54 + (48*n^2 - 304*n + 152)*x - (104*n - 136)*x^2 + 32*x^3)*P[n - 1, x] + (39*n - 26 + (138*n - 118)*x + (132*n - 180)*x^2 + (24*n - 104)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(-6*n^2 + 28*n - 22 + (12*n^2 - 4*n - 8)*x - (8*n - 8)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(48*n^2 - 158*n + 54 + (48*n^2 - 304*n + 152)*x - (104*n - 136)*x^2 + 32*x^3)*v[i - 1] + (39*n - 26 + (138*n - 118)*x + (132*n - 180)*x^2 + (24*n - 104)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(-6*n^2 + 28*n - 22 + (12*n^2 - 4*n - 8)*x - (8*n - 8)*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-d5c14b9fec875ecc","steps":["basis A013609","D"],"edges":["e-eac85dfbd4ac8581","e-fa9b743106e3ffc0"],"text":"A121574 \u2192 basis A013609 \u2192 D"},"headline":"A121574 \u2192 basis A013609 \u2192 D","chain":null}