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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":"a8b4b39873bfa36d3ca6485ea9498342975ed6d70ebb9d6c0e688789cdadd2be","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-c856ca922af03076","role":"sequence","name":"Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.","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:     2,        1;\n  2:     6,        7,         1;\n  3:    22,       49,        17,         1;\n  4:    86,      379,       237,        34,         1;\n  5:   342,     3157,      3289,       847,        60,        1;\n  6:  1366,    27391,     47333,     19448,      2457,       97,       1;\n  7:  5462,   242425,    706641,    442013,     87880,     6118,     147,     1;\n  8: 21846,  2165443,  10837789,  10188542,   2990793,   324326,   13566,   212,   1;\n  9: 87382, 19423453, 169139273, 239508699, 100524060, 15864791, 1022618, 27474, 294, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4-3*x-2*x^2+x^3)*P(n-2,x) - (5*x-8*x^2+3*x^3)*d/dx P(n-2,x) - (x^2-2*x^3+x^4)*d^2/dx^2 P(n-2,x) + (5-3*x+x^2)*P(n-1,x) + 5*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-(4 - 3*x - 2*x^2 + x^3)*P[n - 2, x] - (5*x - 8*x^2 + 3*x^3)*D[P[n - 2, x], x] - (x^2 - 2*x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (5 - 3*x + x^2)*P[n - 1, x] + 5*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(4 - 3*x - 2*x^2 + x^3)*v[i - 2] - (5*x - 8*x^2 + 3*x^3)*deriv(v[i - 2], x) - (x^2 - 2*x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (5 - 3*x + x^2)*v[i - 1] + 5*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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-c856ca922af03076","steps":["D+1"],"edges":["e-95efef8d4e332338"],"text":"A036969 \u2192 D+1"},"headline":"A036969 \u2192 D+1","chain":null}