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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":"94301dc5974b15a80e3d7c679229ee1a1e4351cab1fcedbeba17771735a13985","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-eda3c556a6b4857f","role":"sequence","name":"Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.","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: 1,    1;\n   3: 1,    7,      3;\n   4: 1,   21,     55,      15;\n   5: 1,   51,    340,     525,      105;\n   6: 1,  113,   1526,    5530,     5985,       945;\n   7: 1,  239,   5873,   39718,    96075,     79695,     10395;\n   8: 1,  493,  20721,  234745,  1009855,   1812195,   1216215,    135135;\n   9: 1, 1003,  69306, 1234915,  8486170,  26231205,  37207170,  20945925,   2027025;\n  10: 1, 2025, 223960, 6023116, 62097750, 297166870, 709668960, 829728900, 402026625, 34459425;\n","recurrence_sentence":"The row polynomials satisfy (n-3-2*x)*P(n,x) = -((n^3-6*n^2+12*n-8)*x^2-(3*n^2-12*n+12)*x^3+(2*n-4)*x^4)*P(n-2,x) - ((n^2-5*n+6)*x^2+(n^2-8*n+14)*x^3-(3*n-10)*x^4+2*x^5)*d/dx P(n-2,x) + (n-3+(2*n^2-7*n+1)*x-(5*n-6)*x^2+2*x^3)*P(n-1,x) + ((n-3)*x+(n-5)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-((n^3 - 6*n^2 + 12*n - 8)*x^2 - (3*n^2 - 12*n + 12)*x^3 + (2*n - 4)*x^4)*P[n - 2, x] - ((n^2 - 5*n + 6)*x^2 + (n^2 - 8*n + 14)*x^3 - (3*n - 10)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (n - 3 + (2*n^2 - 7*n + 1)*x - (5*n - 6)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 3)*x + (n - 5)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 3 - 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (-((n^3 - 6*n^2 + 12*n - 8)*x^2 - (3*n^2 - 12*n + 12)*x^3 + (2*n - 4)*x^4)*v[i - 2] - ((n^2 - 5*n + 6)*x^2 + (n^2 - 8*n + 14)*x^3 - (3*n - 10)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (n - 3 + (2*n^2 - 7*n + 1)*x - (5*n - 6)*x^2 + 2*x^3)*v[i - 1] + ((n - 3)*x + (n - 5)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 3 - 2*x)); 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-eda3c556a6b4857f","steps":["D+1"],"edges":["e-06838b408243a8c1"],"text":"A269939 \u2192 D+1"},"headline":"A269939 \u2192 D+1","chain":null}