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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":"ec0d037bbd2e86bf93847d4350fc9ad01b31541a7646a5b85d365a50d38c23a4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f72d27cff21cd6ce","role":"sequence","name":"basis_map_A007318 of Coefficients of a special case of Poisson-Charlier polynomials.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       3,        1;\n   3:      10,        7,        1;\n   4:      41,       47,       14,         1;\n   5:     209,      358,      191,        32,         1;\n   6:    1287,     3069,     2587,       887,        99,        1;\n   7:    9270,    29001,    34757,     19587,      5009,      427,        1;\n   8:   76225,   298787,   472719,    382232,    163489,    33770,     2386,       1;\n   9:  703337,  3336516,  6621414,   7077284,   4352499,  1511036,   263710,   16088,      1;\n  10: 7190779, 40209321, 96522654, 129322750, 105054261, 52408155, 15379454, 2332566, 125691, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-4)*P(n,x) = -(3*n^3-26*n^2+81*n-90+(10*n^3-94*n^2+298*n-318)*x+(12*n^3-120*n^2+384*n-396)*x^2+(6*n^3-62*n^2+198*n-198)*x^3+(n^3-10*n^2+31*n-30)*x^4)*P(n-3,x) - (n^3-8*n^2+23*n-24+(4*n^3-34*n^2+100*n-102)*x+(6*n^3-54*n^2+162*n-162)*x^2+(4*n^3-38*n^2+116*n-114)*x^3+(n^3-10*n^2+31*n-30)*x^4)*d/dx P(n-3,x) + (n^3-12*n^2+39*n-38+(3*n^3-33*n^2+100*n-90)*x+(3*n^3-30*n^2+83*n-70)*x^2+(n^3-9*n^2+24*n-20)*x^3)*P(n-2,x) + (n^3-6*n^2+13*n-8+(3*n^3-21*n^2+48*n-34)*x+(3*n^3-24*n^2+59*n-46)*x^2+(n^3-9*n^2+24*n-20)*x^3)*d/dx P(n-2,x) + (3*n^2-7*n+2+(2*n^2-4*n-2)*x-(n^2-7*n+10)*x^2)*P(n-1,x) - (n^2-5*n+8+(2*n^2-12*n+18)*x+(n^2-7*n+10)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 3 + x, P(3,x) = 10 + 7*x + x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + x;\nP[3, x_] := 10 + 7*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n^3 - 26*n^2 + 81*n - 90 + (10*n^3 - 94*n^2 + 298*n - 318)*x + (12*n^3 - 120*n^2 + 384*n - 396)*x^2 + (6*n^3 - 62*n^2 + 198*n - 198)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (4*n^3 - 34*n^2 + 100*n - 102)*x + (6*n^3 - 54*n^2 + 162*n - 162)*x^2 + (4*n^3 - 38*n^2 + 116*n - 114)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*D[P[n - 3, x], x] + (n^3 - 12*n^2 + 39*n - 38 + (3*n^3 - 33*n^2 + 100*n - 90)*x + (3*n^3 - 30*n^2 + 83*n - 70)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (3*n^3 - 21*n^2 + 48*n - 34)*x + (3*n^3 - 24*n^2 + 59*n - 46)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*D[P[n - 2, x], x] + (3*n^2 - 7*n + 2 + (2*n^2 - 4*n - 2)*x - (n^2 - 7*n + 10)*x^2)*P[n - 1, x] - (n^2 - 5*n + 8 + (2*n^2 - 12*n + 18)*x + (n^2 - 7*n + 10)*x^2)*D[P[n - 1, x], x])/(2*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; v[3] = 10 + 7*x + x^2; for(n = 4, N, my(i = n); v[i] = (-(3*n^3 - 26*n^2 + 81*n - 90 + (10*n^3 - 94*n^2 + 298*n - 318)*x + (12*n^3 - 120*n^2 + 384*n - 396)*x^2 + (6*n^3 - 62*n^2 + 198*n - 198)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (4*n^3 - 34*n^2 + 100*n - 102)*x + (6*n^3 - 54*n^2 + 162*n - 162)*x^2 + (4*n^3 - 38*n^2 + 116*n - 114)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*deriv(v[i - 3], x) + (n^3 - 12*n^2 + 39*n - 38 + (3*n^3 - 33*n^2 + 100*n - 90)*x + (3*n^3 - 30*n^2 + 83*n - 70)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (3*n^3 - 21*n^2 + 48*n - 34)*x + (3*n^3 - 24*n^2 + 59*n - 46)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*deriv(v[i - 2], x) + (3*n^2 - 7*n + 2 + (2*n^2 - 4*n - 2)*x - (n^2 - 7*n + 10)*x^2)*v[i - 1] - (n^2 - 5*n + 8 + (2*n^2 - 12*n + 18)*x + (n^2 - 7*n + 10)*x^2)*deriv(v[i - 1], x))/(2*n - 4)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c9dce8c25621fdea","steps":["x \u21a6 x+1","D+1"],"edges":["e-0491ee9eb3cdd970","e-70eeac67c5be4e95"],"text":"A046716 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A046716 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}