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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":"3a7dc0b43089cb5e3cba38eba6edb3463a9d0ce418ad9b485dc63c7150406b4d","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-a7d7f0f34831fe03","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(2*n+1, 2*k+1)*binomial(2*n-2*k, n-k)/(n+1-k) for 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:       4,        1;\n   3:      20,       12,        1;\n   4:     105,      112,       24,        1;\n   5:     546,      924,      360,       40,       1;\n   6:    2772,     6930,     4290,      880,      60,       1;\n   7:   13728,    48048,    43758,    14300,    1820,      84,      1;\n   8:   66495,   312312,   396396,   190190,   38675,    3360,    112,    1;\n   9:  316030,  1925352,  3267264,  2178176,  649740,   90440,   5712,  144,   1;\n  10: 1478048, 11362494, 24942060, 22170720, 9170616, 1872108, 189924, 9120, 180, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^4-5*n^3+13*n^2+8*n+(4*n^3-14*n^2-8*n)*x+(2*n^2+n)*x^2)*P(n,x) = (8*n^4-8*n^3+46*n^2+2*n-12+(6*n^4-6*n^3-28*n^2-28*n+20)*x+(12*n^3-50*n^2+60*n-16)*x^2+(6*n^2-8*n+2)*x^3)*P(n-1,x) + ((16*n^3-28*n^2+44*n-8)*x-(4*n^3-39*n^2+115*n-26)*x^2-(8*n^2-42*n+10)*x^3-(4*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[((8*n^4 - 8*n^3 + 46*n^2 + 2*n - 12 + (6*n^4 - 6*n^3 - 28*n^2 - 28*n + 20)*x + (12*n^3 - 50*n^2 + 60*n - 16)*x^2 + (6*n^2 - 8*n + 2)*x^3)*P[n - 1, x] + ((16*n^3 - 28*n^2 + 44*n - 8)*x - (4*n^3 - 39*n^2 + 115*n - 26)*x^2 - (8*n^2 - 42*n + 10)*x^3 - (4*n - 1)*x^4)*D[P[n - 1, x], x])/(2*n^4 - 5*n^3 + 13*n^2 + 8*n + (4*n^3 - 14*n^2 - 8*n)*x + (2*n^2 + n)*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] = ((8*n^4 - 8*n^3 + 46*n^2 + 2*n - 12 + (6*n^4 - 6*n^3 - 28*n^2 - 28*n + 20)*x + (12*n^3 - 50*n^2 + 60*n - 16)*x^2 + (6*n^2 - 8*n + 2)*x^3)*v[i - 1] + ((16*n^3 - 28*n^2 + 44*n - 8)*x - (4*n^3 - 39*n^2 + 115*n - 26)*x^2 - (8*n^2 - 42*n + 10)*x^3 - (4*n - 1)*x^4)*deriv(v[i - 1], x))/(2*n^4 - 5*n^3 + 13*n^2 + 8*n + (4*n^3 - 14*n^2 - 8*n)*x + (2*n^2 + n)*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-a7d7f0f34831fe03","steps":["D+1"],"edges":["e-f6b11f38ee1483cf"],"text":"A281000 \u2192 D+1"},"headline":"A281000 \u2192 D+1","chain":null}