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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":"396237f531fca3788812b3fe6bb2e9981d070d2a7822b481de7598d5c3b180d7","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-2c8bb380cc75972b","role":"sequence","name":"Triangle related to the asymptotic expansion of E(x,m=3,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:        0;\n  1:        3;\n  2:       18,        12;\n  3:      105,       120,        30;\n  4:      675,      1020,       450,       60;\n  5:     4872,      8820,      5250,     1260,      105;\n  6:    39396,     81228,     58800,    19320,     2940,     168;\n  7:   354372,    807408,    673470,   272160,    57330,    6048,    252;\n  8:  3518100,   8684160,   8079750,  3796380,   992250,  146160,  11340,   360;\n  9: 38260728, 100914000, 102507900, 54123300, 16566165, 3049200, 332640, 19800, 495;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-8*n^3+24*n^2-32*n+16+(4*n^3-24*n^2+48*n-32)*x+(6*n^2-24*n+24)*x^2+(4*n-8)*x^3+x^4)*P(n-4,x) + (4*n^3-18*n^2+28*n-15+(12*n^2-36*n+28)*x+(12*n-18)*x^2+4*x^3)*P(n-3,x) - (6*n^2-12*n+7+(12*n-12)*x+6*x^2)*P(n-2,x) + (4*n-2+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 3, P(2,x) = 18 + 12*x, P(3,x) = 105 + 120*x + 30*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[2, x_] := 18 + 12*x;\nP[3, x_] := 105 + 120*x + 30*x^2;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^3 - 24*n^2 + 48*n - 32)*x + (6*n^2 - 24*n + 24)*x^2 + (4*n - 8)*x^3 + x^4)*P[n - 4, x] + (4*n^3 - 18*n^2 + 28*n - 15 + (12*n^2 - 36*n + 28)*x + (12*n - 18)*x^2 + 4*x^3)*P[n - 3, x] - (6*n^2 - 12*n + 7 + (12*n - 12)*x + 6*x^2)*P[n - 2, x] + (4*n - 2 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; v[3] = 18 + 12*x; v[4] = 105 + 120*x + 30*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^3 - 24*n^2 + 48*n - 32)*x + (6*n^2 - 24*n + 24)*x^2 + (4*n - 8)*x^3 + x^4)*v[i - 4] + (4*n^3 - 18*n^2 + 28*n - 15 + (12*n^2 - 36*n + 28)*x + (12*n - 18)*x^2 + 4*x^3)*v[i - 3] - (6*n^2 - 12*n + 7 + (12*n - 12)*x + 6*x^2)*v[i - 2] + (4*n - 2 + 4*x)*v[i - 1]); 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-efe23a76953cf69d","steps":["D\u00b2\u2218x (shift +1)","D"],"edges":["e-337faa044bd16e1a","e-d8db31aa64d87a27"],"text":"A130534 \u2192 D\u00b2\u2218x (shift +1) \u2192 D"},"headline":"A130534 \u2192 D\u00b2\u2218x (shift +1) \u2192 D","chain":null}