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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":"2efe1e55154d0f05671a8bd55878611b100a928bc9043f606f227f813ece6d10","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1e2f1b7f566b371c","role":"sequence","name":"Pascal-(1,7,1) array.","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:  1;\n  1:  2,    1;\n  2: 10,   11,     1;\n  3: 18,   51,    20,      1;\n  4: 26,  219,   172,     29,      1;\n  5: 34,  515,   964,    373,     38,      1;\n  6: 42,  939,  3932,   2957,    654,     47,     1;\n  7: 50, 1491, 10612,  16485,   6902,   1015,    56,    1;\n  8: 58, 2171, 22540,  66045,  50526,  13503,  1456,   65,  1;\n  9: 66, 2979, 41252, 193109, 269766, 124551, 23464, 1977, 74, 1;\n","recurrence_sentence":"The row polynomials satisfy -(8*n^3-22*n^2+14*n-(8*n^3-35*n^2+72*n)*x-(28*n^2-69*n)*x^2-(n^2+26*n)*x^3-n*x^4)*P(n,x) = -(8*n^3-14*n^2+6*n+(120*n^3-429*n^2+219*n)*x+(412*n^2-1034*n+659)*x^2+(15*n^2+363*n-391)*x^3+(14*n-43)*x^4-x^5)*P(n-1,x) + ((8*n^2-6*n)*x+(240*n^2-151*n-14)*x^2+(8*n^2+865*n-391)*x^3+(59*n+857)*x^4+(n+59)*x^5+x^6)*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[(-(8*n^3 - 14*n^2 + 6*n + (120*n^3 - 429*n^2 + 219*n)*x + (412*n^2 - 1034*n + 659)*x^2 + (15*n^2 + 363*n - 391)*x^3 + (14*n - 43)*x^4 - x^5)*P[n - 1, x] + ((8*n^2 - 6*n)*x + (240*n^2 - 151*n - 14)*x^2 + (8*n^2 + 865*n - 391)*x^3 + (59*n + 857)*x^4 + (n + 59)*x^5 + x^6)*D[P[n - 1, x], x])/(-8*n^3 + 22*n^2 - 14*n + (8*n^3 - 35*n^2 + 72*n)*x + (28*n^2 - 69*n)*x^2 + (n^2 + 26*n)*x^3 + n*x^4)]];\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] = (-(8*n^3 - 14*n^2 + 6*n + (120*n^3 - 429*n^2 + 219*n)*x + (412*n^2 - 1034*n + 659)*x^2 + (15*n^2 + 363*n - 391)*x^3 + (14*n - 43)*x^4 - x^5)*v[i - 1] + ((8*n^2 - 6*n)*x + (240*n^2 - 151*n - 14)*x^2 + (8*n^2 + 865*n - 391)*x^3 + (59*n + 857)*x^4 + (n + 59)*x^5 + x^6)*deriv(v[i - 1], x))/(-8*n^3 + 22*n^2 - 14*n + (8*n^3 - 35*n^2 + 72*n)*x + (28*n^2 - 69*n)*x^2 + (n^2 + 26*n)*x^3 + n*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-1e2f1b7f566b371c","steps":["D+1"],"edges":["e-e37cbf39b02138a5"],"text":"A081582 \u2192 D+1"},"headline":"A081582 \u2192 D+1","chain":null}