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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":"a91dd27269eb8031411c95deae17f08884654f7d8cb04c0d9b48ef967bc6f0b2","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-dfe248045567f938","role":"sequence","name":"basis_map_A013609 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","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:        7,         4;\n   3:       49,        84,         24;\n   4:      327,       984,        840,        160;\n   5:     2121,      9240,      13800,       7840,       1120;\n   6:    13503,     77340,     167160,     165760,      70560,       8064;\n   7:    84841,    603372,    1715280,    2459520,    1824480,     620928,      59136;\n   8:   527887,   4489072,   15854832,   29897280,   32044320,   18982656,    5381376,     439296;\n   9:  3259921,  32276592,  136387440,  319556160,  450283680,  385152768,  189885696,   46126080,   3294720;\n  10: 20011383, 226163700, 1113258960, 3121345920, 5464499040, 6142840704, 4370446080, 1845043200, 392071680, 24893440;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-8*n^2+12*n-6+(n-1)*x-(2*n-2)*x^2)*P(n,x) = (6*n^3-12*n^2+8*n-2+(8*n^3-16*n^2+7*n+1)*x-(10*n-10)*x^2-(8*n-8)*x^3)*P(n-1,x) + (4*n^2-8*n+2+(12*n^2-24*n)*x+(8*n^2-16*n-18)*x^2-24*x^3-8*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[((6*n^3 - 12*n^2 + 8*n - 2 + (8*n^3 - 16*n^2 + 7*n + 1)*x - (10*n - 10)*x^2 - (8*n - 8)*x^3)*P[n - 1, x] + (4*n^2 - 8*n + 2 + (12*n^2 - 24*n)*x + (8*n^2 - 16*n - 18)*x^2 - 24*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - 8*n^2 + 12*n - 6 + (n - 1)*x - (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] = ((6*n^3 - 12*n^2 + 8*n - 2 + (8*n^3 - 16*n^2 + 7*n + 1)*x - (10*n - 10)*x^2 - (8*n - 8)*x^3)*v[i - 1] + (4*n^2 - 8*n + 2 + (12*n^2 - 24*n)*x + (8*n^2 - 16*n - 18)*x^2 - 24*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - 8*n^2 + 12*n - 6 + (n - 1)*x - (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-4689fb1386d440ad","steps":["basis A013609","D+1"],"edges":["e-7c694c7111620c7d","e-ca34543397b0b1a1"],"text":"A063007 \u2192 basis A013609 \u2192 D+1"},"headline":"A063007 \u2192 basis A013609 \u2192 D+1","chain":null}