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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":"937dfb4a0540e3683796d75b73cc143e2c926856f3df6e99dab49cb6bcbd2531","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f00f6de2d4bbc6f5","role":"sequence","name":"Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       4,        1;\n   3:      20,       10,        1;\n   4:     110,       85,       18,       1;\n   5:     637,      676,      228,      28,       1;\n   6:    3808,     5180,     2450,     490,      40,      1;\n   7:   23256,    38760,    23868,    6800,     920,     54,     1;\n   8:  144210,   285285,   217854,   81795,   15960,   1575,    70,    1;\n   9:  904475,  2074600,  1898512,  892584,  231770,  33320,  2520,   88,   1;\n  10: 5722860, 14948505, 15984540, 9077640, 2975280, 573804, 63756, 3828, 108, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4+2*n^3-4*n^2-2*n+(8*n^3-8*n^2-6*n)*x+(4*n^2+2*n)*x^2)*P(n,x) = (27*n^4-75*n^2+60*n-12+(12*n^4+46*n^3-159*n^2+121*n-26)*x+(24*n^3-29*n^2+13*n-2)*x^2+(12*n^2-16*n+4)*x^3)*P(n-1,x) - ((9*n^3+9*n^2-16*n+4)*x+(8*n^3+28*n^2-32*n+8)*x^2+(16*n^2-3*n-1)*x^3+(8*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((27*n^4 - 75*n^2 + 60*n - 12 + (12*n^4 + 46*n^3 - 159*n^2 + 121*n - 26)*x + (24*n^3 - 29*n^2 + 13*n - 2)*x^2 + (12*n^2 - 16*n + 4)*x^3)*P[n - 1, x] - ((9*n^3 + 9*n^2 - 16*n + 4)*x + (8*n^3 + 28*n^2 - 32*n + 8)*x^2 + (16*n^2 - 3*n - 1)*x^3 + (8*n - 2)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 2*n^3 - 4*n^2 - 2*n + (8*n^3 - 8*n^2 - 6*n)*x + (4*n^2 + 2*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((27*n^4 - 75*n^2 + 60*n - 12 + (12*n^4 + 46*n^3 - 159*n^2 + 121*n - 26)*x + (24*n^3 - 29*n^2 + 13*n - 2)*x^2 + (12*n^2 - 16*n + 4)*x^3)*v[i - 1] - ((9*n^3 + 9*n^2 - 16*n + 4)*x + (8*n^3 + 28*n^2 - 32*n + 8)*x^2 + (16*n^2 - 3*n - 1)*x^3 + (8*n - 2)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 2*n^3 - 4*n^2 - 2*n + (8*n^3 - 8*n^2 - 6*n)*x + (4*n^2 + 2*n)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f00f6de2d4bbc6f5","steps":["D+1"],"edges":["e-2c31379bbd6d00cf"],"text":"A243662 \u2192 D+1"},"headline":"A243662 \u2192 D+1","chain":null}