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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":"ad2bca7a1315eb5f4573fde24def5e131c741eb9dde1122a683b9689694902d1","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-5a70a6e97131d437","role":"sequence","name":"Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","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:  3,   4;\n  1:  5,  16,   12;\n  2:  6,  26,   36,   16;\n  3:  8,  50,  114,  112,    40;\n  4:  9,  66,  189,  264,   180,    48;\n  5: 11, 102,  387,  768,   840,   480,   112;\n  6: 12, 124,  540, 1284,  1800,  1488,   672,   128;\n  7: 14, 172,  912, 2724,  5010,  5808,  4144,  1664,   288;\n  8: 15, 200, 1170, 3940,  8415, 11820, 10920,  6400,  2160,  320;\n  9: 17, 260, 1770, 7060, 18265, 32016, 38500, 31360, 16560, 5120, 704;\n","recurrence_sentence":"The row polynomials satisfy (9*n^2+13*n+4+(24*n^2+28*n+4)*x+(16*n^2+16*n)*x^2)*P(n,x) = (27*n^2+53*n+6+(27*n^3+204*n^2+283*n+14)*x+(108*n^3+512*n^2+548*n+8)*x^2+(144*n^3+528*n^2+448*n)*x^3+(64*n^3+192*n^2+128*n)*x^4)*P(n-1,x) - (12*n+20+(18*n^2+130*n+160)*x+(102*n^2+482*n+484)*x^2+(212*n^2+812*n+696)*x^3+(192*n^2+640*n+480)*x^4+(64*n^2+192*n+128)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3 + 4*x.","mathematica":"P[0, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((27*n^2 + 53*n + 6 + (27*n^3 + 204*n^2 + 283*n + 14)*x + (108*n^3 + 512*n^2 + 548*n + 8)*x^2 + (144*n^3 + 528*n^2 + 448*n)*x^3 + (64*n^3 + 192*n^2 + 128*n)*x^4)*P[n - 1, x] - (12*n + 20 + (18*n^2 + 130*n + 160)*x + (102*n^2 + 482*n + 484)*x^2 + (212*n^2 + 812*n + 696)*x^3 + (192*n^2 + 640*n + 480)*x^4 + (64*n^2 + 192*n + 128)*x^5)*D[P[n - 1, x], x])/(9*n^2 + 13*n + 4 + (24*n^2 + 28*n + 4)*x + (16*n^2 + 16*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((27*n^2 + 53*n + 6 + (27*n^3 + 204*n^2 + 283*n + 14)*x + (108*n^3 + 512*n^2 + 548*n + 8)*x^2 + (144*n^3 + 528*n^2 + 448*n)*x^3 + (64*n^3 + 192*n^2 + 128*n)*x^4)*v[i - 1] - (12*n + 20 + (18*n^2 + 130*n + 160)*x + (102*n^2 + 482*n + 484)*x^2 + (212*n^2 + 812*n + 696)*x^3 + (192*n^2 + 640*n + 480)*x^4 + (64*n^2 + 192*n + 128)*x^5)*deriv(v[i - 1], x))/(9*n^2 + 13*n + 4 + (24*n^2 + 28*n + 4)*x + (16*n^2 + 16*n)*x^2)); 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-5a70a6e97131d437","steps":["D"],"edges":["e-4ace2775516c13f5"],"text":"A064861 \u2192 D"},"headline":"A064861 \u2192 D","chain":null}