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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":"e5f13b3b7d4aea7482ea0259d192f595370f9cc6fd18d8c6c687e7a4b39fd88a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-d5c14b9fec875ecc","role":"sequence","name":"Riordan array (1/(1-2*x), x*(1+x)/(1-2*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:    3,     1;\n  2:    9,     7,     1;\n  3:   24,    32,    11,     1;\n  4:   60,   118,    70,    15,     1;\n  5:  144,   380,   335,   123,    19,     1;\n  6:  336,  1120,  1339,   729,   191,    23,    1;\n  7:  768,  3104,  4736,  3504,  1354,   274,   27,   1;\n  8: 1728,  8224, 15352, 14556,  7631,  2264,  372,  31,  1;\n  9: 3840, 21056, 46640, 54360, 36547, 14665, 3513, 485, 35, 1;\n","recurrence_sentence":"The row polynomials satisfy -(6*n^3-6*n^2-(3*n^3-11*n^2+28*n)*x-(5*n^2-15*n)*x^2-(n^2+3*n)*x^3-n*x^4)*P(n,x) = -(12*n^3-12*n+(12*n^3-2*n^2-70*n)*x+(23*n^2-41*n+28)*x^2+(4*n^2+14*n-20)*x^3+(3*n-10)*x^4-x^5)*P(n-1,x) + ((12*n^2+12*n)*x+(24*n^2+52*n)*x^2+(3*n^2+63*n+28)*x^3+(15*n+60)*x^4+(n+15)*x^5+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[(-(12*n^3 - 12*n + (12*n^3 - 2*n^2 - 70*n)*x + (23*n^2 - 41*n + 28)*x^2 + (4*n^2 + 14*n - 20)*x^3 + (3*n - 10)*x^4 - x^5)*P[n - 1, x] + ((12*n^2 + 12*n)*x + (24*n^2 + 52*n)*x^2 + (3*n^2 + 63*n + 28)*x^3 + (15*n + 60)*x^4 + (n + 15)*x^5 + x^6)*D[P[n - 1, x], x])/(-6*n^3 + 6*n^2 + (3*n^3 - 11*n^2 + 28*n)*x + (5*n^2 - 15*n)*x^2 + (n^2 + 3*n)*x^3 + 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] = (-(12*n^3 - 12*n + (12*n^3 - 2*n^2 - 70*n)*x + (23*n^2 - 41*n + 28)*x^2 + (4*n^2 + 14*n - 20)*x^3 + (3*n - 10)*x^4 - x^5)*v[i - 1] + ((12*n^2 + 12*n)*x + (24*n^2 + 52*n)*x^2 + (3*n^2 + 63*n + 28)*x^3 + (15*n + 60)*x^4 + (n + 15)*x^5 + x^6)*deriv(v[i - 1], x))/(-6*n^3 + 6*n^2 + (3*n^3 - 11*n^2 + 28*n)*x + (5*n^2 - 15*n)*x^2 + (n^2 + 3*n)*x^3 + 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-d5c14b9fec875ecc","steps":["D+1"],"edges":["e-b8ffc6ab03d37d48"],"text":"A121574 \u2192 D+1"},"headline":"A121574 \u2192 D+1","chain":null}