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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":"2140a6fe6a3407e9cc4f390bdc80b1ac0549a4b4e05f1cea9b7a66f73b2db851","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-aff68976ca55ced5","role":"sequence","name":"derivative of Square the entries of Pascal's triangle.","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:    0;\n  1:    1;\n  2:    6,     2;\n  3:   27,    24,      3;\n  4:   88,   168,     60,      4;\n  5:  225,   800,    600,    120,      5;\n  6:  486,  2850,   3900,   1620,    210,     6;\n  7:  931,  8232,  18375,  13720,   3675,   336,     7;\n  8: 1632, 20384,  68208,  82320,  39200,  7392,   504,   8;\n  9: 2673, 44928, 211680, 381024, 291060, 96768, 13608, 720, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^4-6*n^3+15*n^2-16*n+6+(2*n^3-10*n^2+14*n-6)*x+(n^2-2*n+1)*x^2)*P(n,x) = (n^4-2*n^3+3*n^2+(3*n^4-14*n^3+26*n^2-28*n+8)*x+(6*n^3-26*n^2+34*n-10)*x^2+(3*n^2-6*n+2)*x^3)*P(n-1,x) + ((2*n^3-6*n^2+8*n-3)*x-(2*n^3-10*n^2+18*n-7)*x^2-(4*n^2-12*n+5)*x^3-(2*n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^4 - 2*n^3 + 3*n^2 + (3*n^4 - 14*n^3 + 26*n^2 - 28*n + 8)*x + (6*n^3 - 26*n^2 + 34*n - 10)*x^2 + (3*n^2 - 6*n + 2)*x^3)*P[n - 1, x] + ((2*n^3 - 6*n^2 + 8*n - 3)*x - (2*n^3 - 10*n^2 + 18*n - 7)*x^2 - (4*n^2 - 12*n + 5)*x^3 - (2*n - 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 6*n^3 + 15*n^2 - 16*n + 6 + (2*n^3 - 10*n^2 + 14*n - 6)*x + (n^2 - 2*n + 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^4 - 2*n^3 + 3*n^2 + (3*n^4 - 14*n^3 + 26*n^2 - 28*n + 8)*x + (6*n^3 - 26*n^2 + 34*n - 10)*x^2 + (3*n^2 - 6*n + 2)*x^3)*v[i - 1] + ((2*n^3 - 6*n^2 + 8*n - 3)*x - (2*n^3 - 10*n^2 + 18*n - 7)*x^2 - (4*n^2 - 12*n + 5)*x^3 - (2*n - 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 6*n^3 + 15*n^2 - 16*n + 6 + (2*n^3 - 10*n^2 + 14*n - 6)*x + (n^2 - 2*n + 1)*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-b13531ad79e7c629","steps":["D","D+1"],"edges":["e-be8e3a0f5d3685bc","e-4fe2d910407127a1"],"text":"A008459 \u2192 D \u2192 D+1"},"headline":"A008459 \u2192 D \u2192 D+1","chain":null}