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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":"95407eefdeffb6cb13bb618781e231e2dc3989ce42019f62244ac79f17772c61","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-bf08de87964eed6e","role":"sequence","name":"basis_map_A013610 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","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:      1;\n  2:      9,       6;\n  3:     46,      90,       27;\n  4:    215,     720,      594,      108;\n  5:    966,    4620,     6750,     3240,      405;\n  6:   4242,   26460,    56700,    49140,    15795,     1458;\n  7:  18348,  141372,   402570,   521640,   306180,    71442,    5103;\n  8:  78507,  720720,  2569644,  4490640,  3997350,  1714608,  306180,   17496;\n  9: 333190, 3552120, 15243228, 33729696, 40977090, 26943840, 8879220, 1259712, 59049;\n","recurrence_sentence":"The row polynomials satisfy (3*n^4-10*n^3+10*n^2-3*n+(6*n^3-17*n^2+11*n)*x+(3*n^2-3*n)*x^2)*P(n,x) = (12*n^4-34*n^3+24*n^2+4*n-6+(27*n^4-66*n^3+7*n^2+66*n-34)*x+(54*n^3-150*n^2+138*n-42)*x^2+(27*n^2-45*n+18)*x^3)*P(n-1,x) - ((6*n^3-8*n^2+2)*x+(18*n^3-12*n^2-16*n+11)*x^2+(36*n^2-42*n+12)*x^3+(18*n-9)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n^4 - 34*n^3 + 24*n^2 + 4*n - 6 + (27*n^4 - 66*n^3 + 7*n^2 + 66*n - 34)*x + (54*n^3 - 150*n^2 + 138*n - 42)*x^2 + (27*n^2 - 45*n + 18)*x^3)*P[n - 1, x] - ((6*n^3 - 8*n^2 + 2)*x + (18*n^3 - 12*n^2 - 16*n + 11)*x^2 + (36*n^2 - 42*n + 12)*x^3 + (18*n - 9)*x^4)*D[P[n - 1, x], x])/(3*n^4 - 10*n^3 + 10*n^2 - 3*n + (6*n^3 - 17*n^2 + 11*n)*x + (3*n^2 - 3*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((12*n^4 - 34*n^3 + 24*n^2 + 4*n - 6 + (27*n^4 - 66*n^3 + 7*n^2 + 66*n - 34)*x + (54*n^3 - 150*n^2 + 138*n - 42)*x^2 + (27*n^2 - 45*n + 18)*x^3)*v[i - 1] - ((6*n^3 - 8*n^2 + 2)*x + (18*n^3 - 12*n^2 - 16*n + 11)*x^2 + (36*n^2 - 42*n + 12)*x^3 + (18*n - 9)*x^4)*deriv(v[i - 1], x))/(3*n^4 - 10*n^3 + 10*n^2 - 3*n + (6*n^3 - 17*n^2 + 11*n)*x + (3*n^2 - 3*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-b1b19325e0094118","steps":["basis A013610","D+1"],"edges":["e-59715d0ff282a908","e-187cf1a65bc38354"],"text":"A367270 \u2192 basis A013610 \u2192 D+1"},"headline":"A367270 \u2192 basis A013610 \u2192 D+1","chain":null}