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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":"1164b24cc47f7131539bb7f73fe3e8f35dfa5a9635898b197e7a84423bc374e9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-01439b9265fd744f","role":"sequence","name":"basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 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-578711929343af9f","kind":"oeis_match","node_a_id":"n-0506e8505d88de13","node_b_id":null,"fingerprint":null,"oeis_id":"A006527","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A006527","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-deb678460e17bdc6","kind":"oeis_match","node_a_id":"n-0506e8505d88de13","node_b_id":null,"fingerprint":null,"oeis_id":"A167875","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":"A167875","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A006527","relation":"match_row_sums"},{"oeis_id":"A167875","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-0506e8505d88de13","oeis_id":"A167875","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":43,"oeis_terms_available":43,"term_count":200,"raw_size":2027}],"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,     8,    3;\n  3:   -8,     8,   20,     4;\n  4:   -5,   -40,   45,    40,      5;\n  5:   30,   -90,  -90,   150,     70,     6;\n  6:   28,   168, -525,   -56,    385,   112,    7;\n  7: -112,   560,  168, -1848,    392,   840,  168,    8;\n  8: -126,  -672, 3780, -2016,  -4620,  2016, 1638,  240,   9;\n  9:  420, -2940,  840, 14280, -13440, -8400, 6300, 2940, 330, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^4+2*n^3-3*n^2+20*n+(2*n^3+4*n^2-16*n)*x+(n^2+4*n)*x^2)*P(n,x) = -(2*n^4-4*n^3-6*n^2-16*n-16-(3*n^4-3*n^3+9*n^2+13*n-2)*x-(6*n^3-n^2-11*n-4)*x^2-(3*n^2+5*n+2)*x^3)*P(n-1,x) - (4*n^3+2*n^2+10*n+12-(6*n^3-5*n^2+15*n+26)*x+(2*n^3-11*n^2+9*n+22)*x^2+(4*n^2-6*n-10)*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 - 4*n^3 - 6*n^2 - 16*n - 16 - (3*n^4 - 3*n^3 + 9*n^2 + 13*n - 2)*x - (6*n^3 - n^2 - 11*n - 4)*x^2 - (3*n^2 + 5*n + 2)*x^3)*P[n - 1, x] - (4*n^3 + 2*n^2 + 10*n + 12 - (6*n^3 - 5*n^2 + 15*n + 26)*x + (2*n^3 - 11*n^2 + 9*n + 22)*x^2 + (4*n^2 - 6*n - 10)*x^3 + (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 + 2*n^3 - 3*n^2 + 20*n + (2*n^3 + 4*n^2 - 16*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] = (-(2*n^4 - 4*n^3 - 6*n^2 - 16*n - 16 - (3*n^4 - 3*n^3 + 9*n^2 + 13*n - 2)*x - (6*n^3 - n^2 - 11*n - 4)*x^2 - (3*n^2 + 5*n + 2)*x^3)*v[i - 1] - (4*n^3 + 2*n^2 + 10*n + 12 - (6*n^3 - 5*n^2 + 15*n + 26)*x + (2*n^3 - 11*n^2 + 9*n + 22)*x^2 + (4*n^2 - 6*n - 10)*x^3 + (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 + 2*n^3 - 3*n^2 + 20*n + (2*n^3 + 4*n^2 - 16*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-01439b9265fd744f","steps":["D+1"],"edges":["e-2036104bf9bd843b"],"text":"basis-map coefficients of A145596 \u2192 D+1"},"headline":"basis-map coefficients of A145596 \u2192 D+1","chain":null}