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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":"df2e8989bc83dd36136f8b4d2f506a3f97d6215eb8bf90ef4df5291097bfe907","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-73875e607efb6f6a","role":"sequence","name":"basis_map_A007318 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","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:      7,      4;\n  1:     33,     40,      12;\n  2:     84,    146,      84,      16;\n  3:    324,    774,     690,     272,      40;\n  4:    756,   2196,    2541,    1464,     420,      48;\n  5:   2700,   9612,   14211,   11168,    4920,    1152,     112;\n  6:   6048,  24624,   42840,   41284,   23800,    8208,    1568,    128;\n  7:  20736,  98064,  202392,  238092,  174610,   81744,   23856,   3968,   288;\n  8:  45360, 237600,  551880,  746040,  646835,  373020,  143080,  35200,  5040,   320;\n  9: 151632, 894240, 2368440, 3709800, 3805605, 2671464, 1299620, 432640, 94320, 12160, 704;\n","recurrence_sentence":"The row polynomials satisfy (49*n^2+57*n+8+(56*n^2+60*n+4)*x+(16*n^2+16*n)*x^2)*P(n,x) = (343*n^3+1463*n^2+1460*n+28+(931*n^3+3580*n^2+3235*n+30)*x+(924*n^3+3248*n^2+2660*n+8)*x^2+(400*n^3+1296*n^2+960*n)*x^3+(64*n^3+192*n^2+128*n)*x^4)*P(n-1,x) - (588*n^2+2268*n+1968+(1946*n^2+7050*n+5776)*x+(2530*n^2+8678*n+6732)*x^2+(1620*n^2+5292*n+3896)*x^3+(512*n^2+1600*n+1120)*x^4+(64*n^2+192*n+128)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 7 + 4*x.","mathematica":"P[0, x_] := 7 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((343*n^3 + 1463*n^2 + 1460*n + 28 + (931*n^3 + 3580*n^2 + 3235*n + 30)*x + (924*n^3 + 3248*n^2 + 2660*n + 8)*x^2 + (400*n^3 + 1296*n^2 + 960*n)*x^3 + (64*n^3 + 192*n^2 + 128*n)*x^4)*P[n - 1, x] - (588*n^2 + 2268*n + 1968 + (1946*n^2 + 7050*n + 5776)*x + (2530*n^2 + 8678*n + 6732)*x^2 + (1620*n^2 + 5292*n + 3896)*x^3 + (512*n^2 + 1600*n + 1120)*x^4 + (64*n^2 + 192*n + 128)*x^5)*D[P[n - 1, x], x])/(49*n^2 + 57*n + 8 + (56*n^2 + 60*n + 4)*x + (16*n^2 + 16*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 7 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((343*n^3 + 1463*n^2 + 1460*n + 28 + (931*n^3 + 3580*n^2 + 3235*n + 30)*x + (924*n^3 + 3248*n^2 + 2660*n + 8)*x^2 + (400*n^3 + 1296*n^2 + 960*n)*x^3 + (64*n^3 + 192*n^2 + 128*n)*x^4)*v[i - 1] - (588*n^2 + 2268*n + 1968 + (1946*n^2 + 7050*n + 5776)*x + (2530*n^2 + 8678*n + 6732)*x^2 + (1620*n^2 + 5292*n + 3896)*x^3 + (512*n^2 + 1600*n + 1120)*x^4 + (64*n^2 + 192*n + 128)*x^5)*deriv(v[i - 1], x))/(49*n^2 + 57*n + 8 + (56*n^2 + 60*n + 4)*x + (16*n^2 + 16*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-5a70a6e97131d437","steps":["x \u21a6 x+1","D"],"edges":["e-4e787ca2d31dd33d","e-a2be0e695b9baaf2"],"text":"A064861 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A064861 \u2192 x \u21a6 x+1 \u2192 D","chain":null}