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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":"6a8f5d81dc940e1c7b1df873abfdd5ab53d2a2d4d51f6bc699fca159124f026b","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-a893b3f9716e8c07","role":"sequence","name":"shift_x_plus_one of Coefficient triangle of certain polynomials N(4; m,x).","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:      11,        5;\n  2:      70,       72,       15;\n  3:     372,      588,      273,       35;\n  4:    1815,     3840,     2772,      784,       70;\n  5:    8437,    22275,    21780,     9660,     1890,     126;\n  6:   38038,   120120,   148005,    89760,    27720,    4032,     210;\n  7:  167960,   616616,   915915,   705705,   300300,   69300,    7854,    330;\n  8:  730626,  3055104,  5309304,  4964960,  2702700,  864864,  156156,  14256,   495;\n  9: 3142790, 14738490, 29325816, 32264232, 21441420, 8828820, 2222220, 324324, 24453, 715;\n","recurrence_sentence":"The row polynomials satisfy (n^4+8*n^3+15*n^2-4*n+(2*n^3+10*n^2+8*n)*x+(n^2+4*n)*x^2)*P(n,x) = (4*n^4+34*n^3+82*n^2+76*n+24+(3*n^4+32*n^3+95*n^2+130*n+60)*x+(6*n^3+33*n^2+68*n+48)*x^2+(3*n^2+10*n+12)*x^3)*P(n-1,x) - ((2*n^3+15*n^2+26*n+12)*x+(2*n^3+19*n^2+42*n+28)*x^2+(4*n^2+18*n+20)*x^3+(2*n+4)*x^4)*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[((4*n^4 + 34*n^3 + 82*n^2 + 76*n + 24 + (3*n^4 + 32*n^3 + 95*n^2 + 130*n + 60)*x + (6*n^3 + 33*n^2 + 68*n + 48)*x^2 + (3*n^2 + 10*n + 12)*x^3)*P[n - 1, x] - ((2*n^3 + 15*n^2 + 26*n + 12)*x + (2*n^3 + 19*n^2 + 42*n + 28)*x^2 + (4*n^2 + 18*n + 20)*x^3 + (2*n + 4)*x^4)*D[P[n - 1, x], x])/(n^4 + 8*n^3 + 15*n^2 - 4*n + (2*n^3 + 10*n^2 + 8*n)*x + (n^2 + 4*n)*x^2)]];\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] = ((4*n^4 + 34*n^3 + 82*n^2 + 76*n + 24 + (3*n^4 + 32*n^3 + 95*n^2 + 130*n + 60)*x + (6*n^3 + 33*n^2 + 68*n + 48)*x^2 + (3*n^2 + 10*n + 12)*x^3)*v[i - 1] - ((2*n^3 + 15*n^2 + 26*n + 12)*x + (2*n^3 + 19*n^2 + 42*n + 28)*x^2 + (4*n^2 + 18*n + 20)*x^3 + (2*n + 4)*x^4)*deriv(v[i - 1], x))/(n^4 + 8*n^3 + 15*n^2 - 4*n + (2*n^3 + 10*n^2 + 8*n)*x + (n^2 + 4*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-9516b04042fe64c0","steps":["x \u21a6 x+1","D+1"],"edges":["e-06bdf5b2aa1ff3d5","e-63462fd740d162eb"],"text":"A062264 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A062264 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}