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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":"cea42d180feecd83d2e0303f38209da00a566556ff486e817c7e20c5ffad7aa0","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-7ac21ea1b3b7e282","role":"sequence","name":"Pascal-(1,9,1) 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: 12,   13,     1;\n  3: 22,   63,    24,      1;\n  4: 32,  313,   234,     35,      1;\n  5: 42,  763,  1444,    525,     46,      1;\n  6: 52, 1413,  6654,   4715,    936,     57,     1;\n  7: 62, 2263, 18864,  29605,  11426,   1467,    68,    1;\n  8: 72, 3313, 41074, 132195,  95516,  22877,  2118,   79,  1;\n  9: 82, 4563, 76284, 409485, 577206, 244287, 40368, 2889, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy -(10*n^3-28*n^2+18*n-(10*n^3-45*n^2+92*n)*x-(36*n^2-89*n)*x^2-(n^2+34*n)*x^3-n*x^4)*P(n,x) = -(10*n^3-18*n^2+8*n+(190*n^3-695*n^2+391*n)*x+(674*n^2-1688*n+1063)*x^2+(19*n^2+611*n-647)*x^3+(18*n-55)*x^4-x^5)*P(n-1,x) + ((10*n^2-8*n)*x+(380*n^2-267*n-18)*x^2+(10*n^2+1399*n-647)*x^3+(75*n+1389)*x^4+(n+75)*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[(-(10*n^3 - 18*n^2 + 8*n + (190*n^3 - 695*n^2 + 391*n)*x + (674*n^2 - 1688*n + 1063)*x^2 + (19*n^2 + 611*n - 647)*x^3 + (18*n - 55)*x^4 - x^5)*P[n - 1, x] + ((10*n^2 - 8*n)*x + (380*n^2 - 267*n - 18)*x^2 + (10*n^2 + 1399*n - 647)*x^3 + (75*n + 1389)*x^4 + (n + 75)*x^5 + x^6)*D[P[n - 1, x], x])/(-10*n^3 + 28*n^2 - 18*n + (10*n^3 - 45*n^2 + 92*n)*x + (36*n^2 - 89*n)*x^2 + (n^2 + 34*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] = (-(10*n^3 - 18*n^2 + 8*n + (190*n^3 - 695*n^2 + 391*n)*x + (674*n^2 - 1688*n + 1063)*x^2 + (19*n^2 + 611*n - 647)*x^3 + (18*n - 55)*x^4 - x^5)*v[i - 1] + ((10*n^2 - 8*n)*x + (380*n^2 - 267*n - 18)*x^2 + (10*n^2 + 1399*n - 647)*x^3 + (75*n + 1389)*x^4 + (n + 75)*x^5 + x^6)*deriv(v[i - 1], x))/(-10*n^3 + 28*n^2 - 18*n + (10*n^3 - 45*n^2 + 92*n)*x + (36*n^2 - 89*n)*x^2 + (n^2 + 34*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-7ac21ea1b3b7e282","steps":["D+1"],"edges":["e-36e148c47d980dd4"],"text":"A143685 \u2192 D+1"},"headline":"A143685 \u2192 D+1","chain":null}