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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":"8878e5b891acc9bb3cd57449e6adf174577699b797f7aa14f46e5c14653b77c2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-fac4d4a000d744e8","role":"sequence","name":"Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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:     1;\n  1:     3,      1;\n  2:    10,      7,      1;\n  3:    33,     38,     12,      1;\n  4:   105,    173,     93,     18,      1;\n  5:   324,    701,    548,    185,     25,     1;\n  6:   977,   2628,   2724,   1354,    325,    33,    1;\n  7:  2895,   9329,  12102,   8040,   2870,   525,   42,    1;\n  8:  8462,  31811,  49679,  41542,  19955,  5474,  798,   52,  1;\n  9: 24465, 105178, 192336, 194451, 118157, 43875, 9660, 1158, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-4*n^2-10*n+5+(16*n^2-18*n+5)*x+(8*n-4)*x^2)*P(n,x) = (32*n^3-66*n+31+(24*n^3+60*n^2-108*n+32)*x+(48*n^2-14*n-19)*x^2+(24*n-20)*x^3)*P(n-1,x) - (20*n^2+30*n-5+(36*n^2+94*n+6)*x+(16*n^2+96*n+43)*x^2+(32*n+48)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((32*n^3 - 66*n + 31 + (24*n^3 + 60*n^2 - 108*n + 32)*x + (48*n^2 - 14*n - 19)*x^2 + (24*n - 20)*x^3)*P[n - 1, x] - (20*n^2 + 30*n - 5 + (36*n^2 + 94*n + 6)*x + (16*n^2 + 96*n + 43)*x^2 + (32*n + 48)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 4*n^2 - 10*n + 5 + (16*n^2 - 18*n + 5)*x + (8*n - 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((32*n^3 - 66*n + 31 + (24*n^3 + 60*n^2 - 108*n + 32)*x + (48*n^2 - 14*n - 19)*x^2 + (24*n - 20)*x^3)*v[i - 1] - (20*n^2 + 30*n - 5 + (36*n^2 + 94*n + 6)*x + (16*n^2 + 96*n + 43)*x^2 + (32*n + 48)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 4*n^2 - 10*n + 5 + (16*n^2 - 18*n + 5)*x + (8*n - 4)*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-fac4d4a000d744e8","steps":["D+1"],"edges":["e-e3271f37ad24627f"],"text":"A060920 \u2192 D+1"},"headline":"A060920 \u2192 D+1","chain":null}