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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":"5a5476aeb946269a283e584dd7b825391cde4959fae65d06bbf5fd0da8b45485","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-da6c9018f05787b2","role":"sequence","name":"basis_map_A013609 of 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:      5,       2;\n  2:     24,      22,       4;\n  3:    103,     158,      68,       8;\n  4:    419,     918,     664,     184,      16;\n  5:   1644,    4716,    4908,    2280,     464,      32;\n  6:   6289,   22380,   30696,   20536,    6976,    1120,     64;\n  7:  23609,  100490,  171900,  151928,   74224,   19808,   2624,    128;\n  8:  87348,  433210,  890224,  983880,  635216,  243296,  53376,   6016,   256;\n  9: 319447, 1810154, 4349612, 5789352, 4679008, 2366848, 743616, 138368, 13568, 512;\n","recurrence_sentence":"The row polynomials satisfy -(24*n^3-60*n^2-50*n-(48*n^3+8*n^2+112*n)*x-(112*n^2-80*n)*x^2-(32*n^2+64*n)*x^3-32*n*x^4)*P(n,x) = -(192*n^3-216*n^2-524*n+118+(192*n^3-160*n^2-1424*n+380)*x+(352*n^2-864*n+16)*x^2+(128*n^2+64*n-736)*x^3+(64*n-416)*x^4-64*x^5)*P(n-1,x) + (156*n^2+234*n+39+(552*n^2+1244*n+424)*x+(528*n^2+2368*n+1612)*x^2+(96*n^2+1920*n+2752)*x^3+(608*n+2128)*x^4+(64*n+640)*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[(-(192*n^3 - 216*n^2 - 524*n + 118 + (192*n^3 - 160*n^2 - 1424*n + 380)*x + (352*n^2 - 864*n + 16)*x^2 + (128*n^2 + 64*n - 736)*x^3 + (64*n - 416)*x^4 - 64*x^5)*P[n - 1, x] + (156*n^2 + 234*n + 39 + (552*n^2 + 1244*n + 424)*x + (528*n^2 + 2368*n + 1612)*x^2 + (96*n^2 + 1920*n + 2752)*x^3 + (608*n + 2128)*x^4 + (64*n + 640)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(-24*n^3 + 60*n^2 + 50*n + (48*n^3 + 8*n^2 + 112*n)*x + (112*n^2 - 80*n)*x^2 + (32*n^2 + 64*n)*x^3 + 32*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] = (-(192*n^3 - 216*n^2 - 524*n + 118 + (192*n^3 - 160*n^2 - 1424*n + 380)*x + (352*n^2 - 864*n + 16)*x^2 + (128*n^2 + 64*n - 736)*x^3 + (64*n - 416)*x^4 - 64*x^5)*v[i - 1] + (156*n^2 + 234*n + 39 + (552*n^2 + 1244*n + 424)*x + (528*n^2 + 2368*n + 1612)*x^2 + (96*n^2 + 1920*n + 2752)*x^3 + (608*n + 2128)*x^4 + (64*n + 640)*x^5 + 64*x^6)*deriv(v[i - 1], x))/(-24*n^3 + 60*n^2 + 50*n + (48*n^3 + 8*n^2 + 112*n)*x + (112*n^2 - 80*n)*x^2 + (32*n^2 + 64*n)*x^3 + 32*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":["basis A013609","D+1"],"edges":["e-eac85dfbd4ac8581","e-3e130f4116d0fd8a"],"text":"A121574 \u2192 basis A013609 \u2192 D+1"},"headline":"A121574 \u2192 basis A013609 \u2192 D+1","chain":null}