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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":"bd84dddcd459d0b2128a67889efae554581fec6f8cdac13ee22406a82e39c8b2","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-590ad09c6151a6a8","role":"sequence","name":"basis_map_A013611 of Pascal-(1,5,1) array.","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:      6,        4;\n  2:     45,       68,       16;\n  3:    196,      680,      448,        64;\n  4:    909,     4776,     6400,      2496,       256;\n  5:   3762,    27100,    60800,     47040,     12800,      1024;\n  6:  15457,   141148,   444560,    572480,    300800,     62464,      4096;\n  7:  61320,   680336,  2790272,   5208960,   4495360,   1761280,    294912,    16384;\n  8: 240057,  3135120, 15862656,  39178368,  49390080,  31309824,   9707520,  1359872,   65536;\n  9: 924958, 13906932, 83873280, 259393152, 438438912, 405129216, 200392704, 51200000, 6160384, 262144;\n","recurrence_sentence":"The row polynomials satisfy (12*n^2-6*n+(24*n^3-6*n)*x+(80*n^2-88*n)*x^2+(16*n^2+52*n)*x^3+16*n*x^4)*P(n,x) = -(72*n^3-228*n^2+36*n+84+(264*n^3-720*n^2-318*n+642)*x+(640*n^2-1740*n+1076)*x^2+(176*n^2+156*n-708)*x^3+(112*n-480)*x^4-64*x^5)*P(n-1,x) + (36*n^2-9+(288*n^2+120*n-78)*x+(600*n^2+984*n-72)*x^2+(96*n^2+2192*n+908)*x^3+(720*n+2276)*x^4+(64*n+736)*x^5+64*x^6)*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[(-(72*n^3 - 228*n^2 + 36*n + 84 + (264*n^3 - 720*n^2 - 318*n + 642)*x + (640*n^2 - 1740*n + 1076)*x^2 + (176*n^2 + 156*n - 708)*x^3 + (112*n - 480)*x^4 - 64*x^5)*P[n - 1, x] + (36*n^2 - 9 + (288*n^2 + 120*n - 78)*x + (600*n^2 + 984*n - 72)*x^2 + (96*n^2 + 2192*n + 908)*x^3 + (720*n + 2276)*x^4 + (64*n + 736)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(12*n^2 - 6*n + (24*n^3 - 6*n)*x + (80*n^2 - 88*n)*x^2 + (16*n^2 + 52*n)*x^3 + 16*n*x^4)]];\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] = (-(72*n^3 - 228*n^2 + 36*n + 84 + (264*n^3 - 720*n^2 - 318*n + 642)*x + (640*n^2 - 1740*n + 1076)*x^2 + (176*n^2 + 156*n - 708)*x^3 + (112*n - 480)*x^4 - 64*x^5)*v[i - 1] + (36*n^2 - 9 + (288*n^2 + 120*n - 78)*x + (600*n^2 + 984*n - 72)*x^2 + (96*n^2 + 2192*n + 908)*x^3 + (720*n + 2276)*x^4 + (64*n + 736)*x^5 + 64*x^6)*deriv(v[i - 1], x))/(12*n^2 - 6*n + (24*n^3 - 6*n)*x + (80*n^2 - 88*n)*x^2 + (16*n^2 + 52*n)*x^3 + 16*n*x^4)); 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-6ad01dc6e705cdb6","steps":["basis A013611","D+1"],"edges":["e-09b3cf169dcfe004","e-ffa31da804973754"],"text":"A081580 \u2192 basis A013611 \u2192 D+1"},"headline":"A081580 \u2192 basis A013611 \u2192 D+1","chain":null}