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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":"1b0ea65b819c38a8d345776d6cbc92a81a2bffdb30782520cdbf0ea560d99b8f","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-3aeb7159e408f97a","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.","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:   2,    1;\n   3:   5,    6,     1;\n   4:  11,   28,    12,      1;\n   5:  21,  104,    90,     20,      1;\n   6:  36,  315,   500,    220,     30,      1;\n   7:  57,  812,  2178,   1700,    455,     42,     1;\n   8:  85, 1848,  7812,  10010,   4655,    840,    56,    1;\n   9: 121, 3816, 24024,  47432,  35490,  10976,  1428,   72,  1;\n  10: 166, 7293, 65340, 188760, 214032, 104958, 23184, 2280, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n^6-14*n^5+12*n^4+26*n^3-16*n^2-12*n+(4*n^5+2*n^4-84*n^3+38*n^2+40*n)*x-(4*n^4-42*n^3+18*n^2+20*n)*x^2-(4*n^3-2*n^2-2*n)*x^3)*P(n,x) = (4*n^6-2*n^5-6*n^4+26*n^3-10*n^2-12*n+(7*n^6-17*n^5-31*n^4+17*n^3-16*n^2+40*n)*x+(7*n^5-75*n^3+140*n^2-72*n)*x^2-(7*n^4-70*n^3+101*n^2-38*n)*x^3-(7*n^3-11*n^2+4*n)*x^4)*P(n-1,x) + ((12*n^5+22*n^4-80*n^3-50*n^2+96*n)*x+(2*n^5-16*n^4+80*n^3-132*n^2-234*n+120)*x^2+(2*n^4-36*n^3+90*n^2+268*n-144)*x^3-(2*n^3-18*n^2+136*n-60)*x^4-(2*n^2-14*n+6)*x^5)*d/dx P(n-1,x) + ((20*n^4-40*n^3-28*n^2+48*n)*x^2-(5*n^4-30*n^3-3*n^2+148*n-48)*x^3-(5*n^3+19*n^2-182*n+68)*x^4+(5*n^2-57*n+22)*x^5+(5*n-2)*x^6)*d^2/dx^2 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[((4*n^6 - 2*n^5 - 6*n^4 + 26*n^3 - 10*n^2 - 12*n + (7*n^6 - 17*n^5 - 31*n^4 + 17*n^3 - 16*n^2 + 40*n)*x + (7*n^5 - 75*n^3 + 140*n^2 - 72*n)*x^2 - (7*n^4 - 70*n^3 + 101*n^2 - 38*n)*x^3 - (7*n^3 - 11*n^2 + 4*n)*x^4)*P[n - 1, x] + ((12*n^5 + 22*n^4 - 80*n^3 - 50*n^2 + 96*n)*x + (2*n^5 - 16*n^4 + 80*n^3 - 132*n^2 - 234*n + 120)*x^2 + (2*n^4 - 36*n^3 + 90*n^2 + 268*n - 144)*x^3 - (2*n^3 - 18*n^2 + 136*n - 60)*x^4 - (2*n^2 - 14*n + 6)*x^5)*D[P[n - 1, x], x] + ((20*n^4 - 40*n^3 - 28*n^2 + 48*n)*x^2 - (5*n^4 - 30*n^3 - 3*n^2 + 148*n - 48)*x^3 - (5*n^3 + 19*n^2 - 182*n + 68)*x^4 + (5*n^2 - 57*n + 22)*x^5 + (5*n - 2)*x^6)*D[P[n - 1, x], {x, 2}])/(4*n^6 - 14*n^5 + 12*n^4 + 26*n^3 - 16*n^2 - 12*n + (4*n^5 + 2*n^4 - 84*n^3 + 38*n^2 + 40*n)*x - (4*n^4 - 42*n^3 + 18*n^2 + 20*n)*x^2 - (4*n^3 - 2*n^2 - 2*n)*x^3)]];\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] = ((4*n^6 - 2*n^5 - 6*n^4 + 26*n^3 - 10*n^2 - 12*n + (7*n^6 - 17*n^5 - 31*n^4 + 17*n^3 - 16*n^2 + 40*n)*x + (7*n^5 - 75*n^3 + 140*n^2 - 72*n)*x^2 - (7*n^4 - 70*n^3 + 101*n^2 - 38*n)*x^3 - (7*n^3 - 11*n^2 + 4*n)*x^4)*v[i - 1] + ((12*n^5 + 22*n^4 - 80*n^3 - 50*n^2 + 96*n)*x + (2*n^5 - 16*n^4 + 80*n^3 - 132*n^2 - 234*n + 120)*x^2 + (2*n^4 - 36*n^3 + 90*n^2 + 268*n - 144)*x^3 - (2*n^3 - 18*n^2 + 136*n - 60)*x^4 - (2*n^2 - 14*n + 6)*x^5)*deriv(v[i - 1], x) + ((20*n^4 - 40*n^3 - 28*n^2 + 48*n)*x^2 - (5*n^4 - 30*n^3 - 3*n^2 + 148*n - 48)*x^3 - (5*n^3 + 19*n^2 - 182*n + 68)*x^4 + (5*n^2 - 57*n + 22)*x^5 + (5*n - 2)*x^6)*deriv(deriv(v[i - 1], x), x))/(4*n^6 - 14*n^5 + 12*n^4 + 26*n^3 - 16*n^2 - 12*n + (4*n^5 + 2*n^4 - 84*n^3 + 38*n^2 + 40*n)*x - (4*n^4 - 42*n^3 + 18*n^2 + 20*n)*x^2 - (4*n^3 - 2*n^2 - 2*n)*x^3)); 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-5c0c5bce3aee8449","steps":["reverse","D+1"],"edges":["e-5d51438768cc4177","e-65b3a58335d7dee2"],"text":"A391045 \u2192 reverse \u2192 D+1"},"headline":"A391045 \u2192 reverse \u2192 D+1","chain":null}