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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":"245b7e932337eb72d67f16453331b0f5d28c330926ef9aace28ae105fa79276b","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:  4,   7,    2;\n  1:  6,  21,   20,    4;\n  2:  7,  32,   49,   28,     4;\n  3:  9,  58,  139,  150,    68,     8;\n  4: 10,  75,  222,  327,   246,    84,     8;\n  5: 12, 113,  438,  897,  1032,   648,   192,    16;\n  6: 13, 136,  602, 1464,  2121,  1848,   920,   224,    16;\n  7: 15, 186,  998, 3028,  5691,  6810,  5112,  2256,   496,   32;\n  8: 16, 215, 1270, 4330,  9400, 13503, 12890,  7960,  2960,  560,   32;\n  9: 18, 277, 1900, 7650, 20030, 35669, 43836, 36860, 20480, 6960, 1216, 64;\n","recurrence_sentence":"The row polynomials satisfy (24*n^2+80*n+34+(32*n^2+176*n+112)*x+(96*n+144)*x^2+64*x^3)*P(n,x) = -(27*n^3-30*n^2-271*n-148+(72*n^3-352*n^2-1490*n-926)*x+(48*n^3-900*n^2-3148*n-2280)*x^2-(768*n^2+3136*n+2656)*x^3-(192*n^2+1472*n+1408)*x^4-(256*n+256)*x^5)*P(n-1,x) + (36*n^2+18*n-154+(204*n^2-26*n-1194)*x+(424*n^2-348*n-3632)*x^2+(384*n^2-592*n-5520)*x^3+(128*n^2-288*n-4400)*x^4-1728*x^5-256*x^6)*d/dx P(n-1,x) - (12*n+28+(104*n+248)*x+(348*n+892)*x^2+(560*n+1664)*x^3+(432*n+1696)*x^4+(128*n+896)*x^5+192*x^6)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 4 + 7*x + 2*x^2.","mathematica":"P[0, x_] := 4 + 7*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(27*n^3 - 30*n^2 - 271*n - 148 + (72*n^3 - 352*n^2 - 1490*n - 926)*x + (48*n^3 - 900*n^2 - 3148*n - 2280)*x^2 - (768*n^2 + 3136*n + 2656)*x^3 - (192*n^2 + 1472*n + 1408)*x^4 - (256*n + 256)*x^5)*P[n - 1, x] + (36*n^2 + 18*n - 154 + (204*n^2 - 26*n - 1194)*x + (424*n^2 - 348*n - 3632)*x^2 + (384*n^2 - 592*n - 5520)*x^3 + (128*n^2 - 288*n - 4400)*x^4 - 1728*x^5 - 256*x^6)*D[P[n - 1, x], x] - (12*n + 28 + (104*n + 248)*x + (348*n + 892)*x^2 + (560*n + 1664)*x^3 + (432*n + 1696)*x^4 + (128*n + 896)*x^5 + 192*x^6)*D[P[n - 1, x], {x, 2}])/(24*n^2 + 80*n + 34 + (32*n^2 + 176*n + 112)*x + (96*n + 144)*x^2 + 64*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4 + 7*x + 2*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(27*n^3 - 30*n^2 - 271*n - 148 + (72*n^3 - 352*n^2 - 1490*n - 926)*x + (48*n^3 - 900*n^2 - 3148*n - 2280)*x^2 - (768*n^2 + 3136*n + 2656)*x^3 - (192*n^2 + 1472*n + 1408)*x^4 - (256*n + 256)*x^5)*v[i - 1] + (36*n^2 + 18*n - 154 + (204*n^2 - 26*n - 1194)*x + (424*n^2 - 348*n - 3632)*x^2 + (384*n^2 - 592*n - 5520)*x^3 + (128*n^2 - 288*n - 4400)*x^4 - 1728*x^5 - 256*x^6)*deriv(v[i - 1], x) - (12*n + 28 + (104*n + 248)*x + (348*n + 892)*x^2 + (560*n + 1664)*x^3 + (432*n + 1696)*x^4 + (128*n + 896)*x^5 + 192*x^6)*deriv(deriv(v[i - 1], x), x))/(24*n^2 + 80*n + 34 + (32*n^2 + 176*n + 112)*x + (96*n + 144)*x^2 + 64*x^3)); 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+1"],"edges":["e-fb87f78803cc76fe"],"text":"A064861 \u2192 D+1"},"headline":"A064861 \u2192 D+1","chain":null}