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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":"4816a46edf82a279295c302fceba9363bda67f2f0f4678078404c7897f27e8a7","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-131f5baf2477adf2","role":"sequence","name":"basis_map_A007318 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).","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:      5,      1;\n  2:     21,     10,      1;\n  3:     85,     66,     15,      1;\n  4:    332,    364,    135,     20,      1;\n  5:   1266,   1807,    949,    228,     25,     1;\n  6:   4741,   8370,   5745,   1952,    345,    30,    1;\n  7:  17507,  36913,  31455,  13978,   3485,   486,   35,   1;\n  8:  63927, 156940, 160243,  88536,  28850,  5660,  651,  40,  1;\n  9: 231301, 648588, 773253, 512778, 207804, 53217, 8589, 840, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy -(3*n^3-15*n^2-23*n+39-(3*n^3+n^2+38*n-36)*x-(7*n^2+3*n+16)*x^2-(n^2+7*n+10)*x^3-(n+1)*x^4)*P(n,x) = -(24*n^3-48*n^2-170*n+214+(12*n^3+4*n^2-260*n+218)*x+(28*n^2-84*n-80)*x^2+(4*n^2+8*n-128)*x^3+(2*n-30)*x^4-2*x^5)*P(n-1,x) + (39*n^2+78*n-39+(69*n^2+242*n+22)*x+(33*n^2+263*n+245)*x^2+(3*n^2+117*n+294)*x^3+(19*n+129)*x^4+(n+20)*x^5+x^6)*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[(-(24*n^3 - 48*n^2 - 170*n + 214 + (12*n^3 + 4*n^2 - 260*n + 218)*x + (28*n^2 - 84*n - 80)*x^2 + (4*n^2 + 8*n - 128)*x^3 + (2*n - 30)*x^4 - 2*x^5)*P[n - 1, x] + (39*n^2 + 78*n - 39 + (69*n^2 + 242*n + 22)*x + (33*n^2 + 263*n + 245)*x^2 + (3*n^2 + 117*n + 294)*x^3 + (19*n + 129)*x^4 + (n + 20)*x^5 + x^6)*D[P[n - 1, x], x])/(-3*n^3 + 15*n^2 + 23*n - 39 + (3*n^3 + n^2 + 38*n - 36)*x + (7*n^2 + 3*n + 16)*x^2 + (n^2 + 7*n + 10)*x^3 + (n + 1)*x^4)]];\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] = (-(24*n^3 - 48*n^2 - 170*n + 214 + (12*n^3 + 4*n^2 - 260*n + 218)*x + (28*n^2 - 84*n - 80)*x^2 + (4*n^2 + 8*n - 128)*x^3 + (2*n - 30)*x^4 - 2*x^5)*v[i - 1] + (39*n^2 + 78*n - 39 + (69*n^2 + 242*n + 22)*x + (33*n^2 + 263*n + 245)*x^2 + (3*n^2 + 117*n + 294)*x^3 + (19*n + 129)*x^4 + (n + 20)*x^5 + x^6)*deriv(v[i - 1], x))/(-3*n^3 + 15*n^2 + 23*n - 39 + (3*n^3 + n^2 + 38*n - 36)*x + (7*n^2 + 3*n + 16)*x^2 + (n^2 + 7*n + 10)*x^3 + (n + 1)*x^4)); 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-d6b7687ba0ff59dd","steps":["basis A007318","D+1"],"edges":["e-32d1efa81dadfdb2","e-c8ff3cde74341dd8"],"text":"A116412 \u2192 basis A007318 \u2192 D+1"},"headline":"A116412 \u2192 basis A007318 \u2192 D+1","chain":null}