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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":"2a01f263f18f408729c44dfe3cc5c71a904e7741cbcf8765ebebc548ced83e96","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-bfe76bb735cd71bb","role":"sequence","name":"basis_map_A007318 of Pascal-(1,3,1) 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:     3,      1;\n  2:    14,      9,      1;\n  3:    50,     54,     15,      1;\n  4:   183,    278,    129,     21,      1;\n  5:   637,   1263,    857,    239,     27,     1;\n  6:  2188,   5367,   4836,   1987,    384,    33,     1;\n  7:  7380,  21628,  24474,  13542,   3868,   564,    39,    1;\n  8: 24605,  83964, 114718,  80510,  31146,  6700,   779,   45,  1;\n  9: 81191, 316461, 507678, 433842, 214302, 62538, 10683, 1029, 51, 1;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2+8*n+(4*n^3+12*n^2+8*n)*x+(15*n^2+7*n)*x^2+(n^2+14*n)*x^3+n*x^4)*P(n,x) = -(32*n^3-8*n^2-120*n+48+(28*n^3+92*n^2-192*n-16)*x+(101*n^2+7*n-198)*x^2+(7*n^2+83*n-157)*x^3+(6*n-24)*x^4-x^5)*P(n-1,x) + (64*n^2+192*n+128+(128*n^2+624*n+624)*x+(68*n^2+700*n+1128)*x^2+(4*n^2+299*n+927)*x^3+(32*n+327)*x^4+(n+33)*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[(-(32*n^3 - 8*n^2 - 120*n + 48 + (28*n^3 + 92*n^2 - 192*n - 16)*x + (101*n^2 + 7*n - 198)*x^2 + (7*n^2 + 83*n - 157)*x^3 + (6*n - 24)*x^4 - x^5)*P[n - 1, x] + (64*n^2 + 192*n + 128 + (128*n^2 + 624*n + 624)*x + (68*n^2 + 700*n + 1128)*x^2 + (4*n^2 + 299*n + 927)*x^3 + (32*n + 327)*x^4 + (n + 33)*x^5 + x^6)*D[P[n - 1, x], x])/(8*n^2 + 8*n + (4*n^3 + 12*n^2 + 8*n)*x + (15*n^2 + 7*n)*x^2 + (n^2 + 14*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] = (-(32*n^3 - 8*n^2 - 120*n + 48 + (28*n^3 + 92*n^2 - 192*n - 16)*x + (101*n^2 + 7*n - 198)*x^2 + (7*n^2 + 83*n - 157)*x^3 + (6*n - 24)*x^4 - x^5)*v[i - 1] + (64*n^2 + 192*n + 128 + (128*n^2 + 624*n + 624)*x + (68*n^2 + 700*n + 1128)*x^2 + (4*n^2 + 299*n + 927)*x^3 + (32*n + 327)*x^4 + (n + 33)*x^5 + x^6)*deriv(v[i - 1], x))/(8*n^2 + 8*n + (4*n^3 + 12*n^2 + 8*n)*x + (15*n^2 + 7*n)*x^2 + (n^2 + 14*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-d8f9ad9ce0e039d0","steps":["x \u21a6 x+1","D+1"],"edges":["e-118ee1d5972a9fb3","e-70309d56d93bb65b"],"text":"A081578 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A081578 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}