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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":"cfdbcacc49114a2c0eb3914da58c11b7f6e0aeeb8d0f61cf05140d2478dd2da3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5d4ff1fdd67b7e1d","role":"sequence","name":"reversal of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","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:    5,     1;\n   3:   20,    12,      1;\n   4:   65,    81,     21,      1;\n   5:  175,   392,    210,     32,      1;\n   6:  406,  1470,   1400,    440,     45,      1;\n   7:  840,  4536,   6930,   3840,    810,     60,     1;\n   8: 1590, 12054,  27342,  24150,   8925,   1365,    77,    1;\n   9: 2805, 28512,  90552, 118272,  69300,  18480,  2156,   96,   1;\n  10: 4675, 61479, 261360, 476784, 411642, 173250, 35112, 3240, 117, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^4-3*n^3+2*n^2+16*n-16+(2*n^3-4*n^2-10*n+12)*x+(n^2+n-2)*x^2)*P(n,x) = (n^4+2*n^3+n^2+12*n+20+(3*n^4-10*n^3+22*n^2-20*n-40)*x+(6*n^3-18*n^2+6*n+24)*x^2+(3*n^2-2*n-4)*x^3)*P(n-1,x) + ((2*n^3-4*n^2+8*n+3)*x-(2*n^3-8*n^2+14*n+7)*x^2-(4*n^2-8*n-5)*x^3-(2*n+1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^4 + 2*n^3 + n^2 + 12*n + 20 + (3*n^4 - 10*n^3 + 22*n^2 - 20*n - 40)*x + (6*n^3 - 18*n^2 + 6*n + 24)*x^2 + (3*n^2 - 2*n - 4)*x^3)*P[n - 1, x] + ((2*n^3 - 4*n^2 + 8*n + 3)*x - (2*n^3 - 8*n^2 + 14*n + 7)*x^2 - (4*n^2 - 8*n - 5)*x^3 - (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 3*n^3 + 2*n^2 + 16*n - 16 + (2*n^3 - 4*n^2 - 10*n + 12)*x + (n^2 + n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^4 + 2*n^3 + n^2 + 12*n + 20 + (3*n^4 - 10*n^3 + 22*n^2 - 20*n - 40)*x + (6*n^3 - 18*n^2 + 6*n + 24)*x^2 + (3*n^2 - 2*n - 4)*x^3)*v[i - 1] + ((2*n^3 - 4*n^2 + 8*n + 3)*x - (2*n^3 - 8*n^2 + 14*n + 7)*x^2 - (4*n^2 - 8*n - 5)*x^3 - (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 3*n^3 + 2*n^2 + 16*n - 16 + (2*n^3 - 4*n^2 - 10*n + 12)*x + (n^2 + n - 2)*x^2)); 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-a2304b34e998691c","steps":["reverse","D+1"],"edges":["e-07ea51e83417b459","e-6bc5c8827f7805b7"],"text":"A062145 \u2192 reverse \u2192 D+1"},"headline":"A062145 \u2192 reverse \u2192 D+1","chain":null}