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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":"39c7e1791355c223803fe631b0f9ec2ded1aa1a971ec9d5eeea09420526be374","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-3bc20048ee6f0787","role":"sequence","name":"basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-ed6b7dfed0c8d29b","kind":"oeis_match","node_a_id":"n-100368592cf6d595","node_b_id":null,"fingerprint":null,"oeis_id":"A006000","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A006000","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A006000","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-100368592cf6d595","oeis_id":"A006000","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":48,"oeis_terms_available":48,"term_count":200,"raw_size":2059}],"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:    2,     2;\n  2:    0,     9,     3;\n  3:  -12,    12,    24,     4;\n  4:  -10,   -60,    70,    50,      5;\n  5:   60,  -180,  -120,   240,     90,      6;\n  6:   70,   315, -1050,    35,    630,    147,     7;\n  7: -280,  1400,     0, -3640,   1120,   1400,   224,    8;\n  8: -378, -1512,  9324, -6300,  -8694,   4788,  2772,  324,   9;\n  9: 1260, -8820,  5040, 33600, -36540, -13860, 14280, 5040, 450, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^4-n^3+12*n+(2*n^3-n^2-10*n)*x+(n^2+2*n)*x^2)*P(n,x) = -(2*n^4-6*n^3-2*n^2-6*n-12-(3*n^4-8*n^3+10*n^2+11*n-10)*x-(6*n^3-8*n^2-12*n+2)*x^2-(3*n^2+3*n)*x^3)*P(n-1,x) - (4*n^3-2*n^2+6*n+12-(6*n^3-11*n^2+13*n+30)*x+(2*n^3-13*n^2+13*n+28)*x^2+(4*n^2-8*n-12)*x^3+(2*n+2)*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[(-(2*n^4 - 6*n^3 - 2*n^2 - 6*n - 12 - (3*n^4 - 8*n^3 + 10*n^2 + 11*n - 10)*x - (6*n^3 - 8*n^2 - 12*n + 2)*x^2 - (3*n^2 + 3*n)*x^3)*P[n - 1, x] - (4*n^3 - 2*n^2 + 6*n + 12 - (6*n^3 - 11*n^2 + 13*n + 30)*x + (2*n^3 - 13*n^2 + 13*n + 28)*x^2 + (4*n^2 - 8*n - 12)*x^3 + (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 - n^3 + 12*n + (2*n^3 - n^2 - 10*n)*x + (n^2 + 2*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] = (-(2*n^4 - 6*n^3 - 2*n^2 - 6*n - 12 - (3*n^4 - 8*n^3 + 10*n^2 + 11*n - 10)*x - (6*n^3 - 8*n^2 - 12*n + 2)*x^2 - (3*n^2 + 3*n)*x^3)*v[i - 1] - (4*n^3 - 2*n^2 + 6*n + 12 - (6*n^3 - 11*n^2 + 13*n + 30)*x + (2*n^3 - 13*n^2 + 13*n + 28)*x^2 + (4*n^2 - 8*n - 12)*x^3 + (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 - n^3 + 12*n + (2*n^3 - n^2 - 10*n)*x + (n^2 + 2*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-3bc20048ee6f0787","steps":["D+1"],"edges":["e-908b120a7c606a06"],"text":"basis-map coefficients of A132812 \u2192 D+1"},"headline":"basis-map coefficients of A132812 \u2192 D+1","chain":null}