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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":"412aabea9bf05441695e9d6085d21c2ba2dbd836eb6d551864e898e1af9519cf","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-1453c6ad660cd835","role":"sequence","name":"reversal of 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:   5,    5,     1;\n  1:  12,   18,     8,     1;\n  2:  16,   38,    31,    10,     1;\n  3:  36,  104,   113,    57,    13,     1;\n  4:  44,  168,   255,   195,    78,    15,     1;\n  5:  96,  416,   744,   708,   384,   117,    18,    1;\n  6: 112,  592,  1328,  1644,  1221,   552,   146,   20,    1;\n  7: 240, 1392,  3496,  4976,  4407,  2505,   906,  198,   23,   1;\n  8: 272, 1840,  5480,  9440, 10385,  7593,  3715, 1190,  235,  25,  1;\n  9: 576, 4192, 13600, 25920, 32180, 27254, 16008, 6485, 1760, 300, 28, 1;\n","recurrence_sentence":"The row polynomials satisfy (9*n^2+31*n+10+(12*n^2+62*n+40)*x+(4*n^2+40*n+50)*x^2+(8*n+24)*x^3+4*x^4)*P(n,x) = -(27*n^3+96*n^2+67*n+34+(54*n^3+265*n^2+281*n+112)*x+(36*n^3+260*n^2+404*n+148)*x^2+(8*n^3+108*n^2+260*n+98)*x^3+(16*n^2+76*n+32)*x^4+(8*n+4)*x^5)*P(n-1,x) + (36*n^2+156*n+152+(102*n^2+530*n+620)*x+(106*n^2+698*n+1016)*x^2+(48*n^2+444*n+852)*x^3+(8*n^2+136*n+384)*x^4+(16*n+88)*x^5+8*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 5 + 5*x + x^2.","mathematica":"P[0, x_] := 5 + 5*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(27*n^3 + 96*n^2 + 67*n + 34 + (54*n^3 + 265*n^2 + 281*n + 112)*x + (36*n^3 + 260*n^2 + 404*n + 148)*x^2 + (8*n^3 + 108*n^2 + 260*n + 98)*x^3 + (16*n^2 + 76*n + 32)*x^4 + (8*n + 4)*x^5)*P[n - 1, x] + (36*n^2 + 156*n + 152 + (102*n^2 + 530*n + 620)*x + (106*n^2 + 698*n + 1016)*x^2 + (48*n^2 + 444*n + 852)*x^3 + (8*n^2 + 136*n + 384)*x^4 + (16*n + 88)*x^5 + 8*x^6)*D[P[n - 1, x], x])/(9*n^2 + 31*n + 10 + (12*n^2 + 62*n + 40)*x + (4*n^2 + 40*n + 50)*x^2 + (8*n + 24)*x^3 + 4*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 5 + 5*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(27*n^3 + 96*n^2 + 67*n + 34 + (54*n^3 + 265*n^2 + 281*n + 112)*x + (36*n^3 + 260*n^2 + 404*n + 148)*x^2 + (8*n^3 + 108*n^2 + 260*n + 98)*x^3 + (16*n^2 + 76*n + 32)*x^4 + (8*n + 4)*x^5)*v[i - 1] + (36*n^2 + 156*n + 152 + (102*n^2 + 530*n + 620)*x + (106*n^2 + 698*n + 1016)*x^2 + (48*n^2 + 444*n + 852)*x^3 + (8*n^2 + 136*n + 384)*x^4 + (16*n + 88)*x^5 + 8*x^6)*deriv(v[i - 1], x))/(9*n^2 + 31*n + 10 + (12*n^2 + 62*n + 40)*x + (4*n^2 + 40*n + 50)*x^2 + (8*n + 24)*x^3 + 4*x^4)); 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":["reverse","D+1"],"edges":["e-75e40465d2024023","e-e1dfe1398a3064d9"],"text":"A064861 \u2192 reverse \u2192 D+1"},"headline":"A064861 \u2192 reverse \u2192 D+1","chain":null}