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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"4998f3dc1357a7a74e2bf545f7234fc4eb09a011170fc1de20ff384d2d26d48d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6a8ffbe72ba42678","role":"sequence","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-x)^4, read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   11,      12,        1;\n   2:   24,      69,       26,         1;\n   3:   51,     326,      288,        54,          1;\n   4:  106,    1333,     2456,      1034,        110,          1;\n   5:  217,    4950,    17103,     14380,       3447,        222,          1;\n   6:  440,   17233,   102846,    157415,      73468,      11031,        446,         1;\n   7:  887,   57414,   557206,   1435634,    1196400,     345898,      34466,       894,        1;\n   8: 1782,  185541,  2801192,  11431240,   15792204,    8050658,    1546264,    106128,     1790,      1;\n   9: 3573,  586910, 13334037,  82275048,  177403938,  148475796,   49914090,   6680808,   323817,   3582,    1;\n  10: 7156, 1828609, 60955262, 548771397, 1761587808, 2281158138, 1251203700, 292307898, 28211388, 982277, 7166, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1+2*x)*P(n,x) = -(2*n+(2*n^2-3*n+4)*x+(n^3-2*n^2+5*n-6)*x^2+(3*n^2-4*n+2)*x^3+2*n*x^4)*P(n-2,x) - (n*x+(n^2-2*n+2)*x^2-(n^2-4*n+4)*x^3-(3*n-4)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n-2+(2*n^2-3*n+7)*x+(5*n-4)*x^2+2*x^3)*P(n-1,x) + ((n-1)*x-(n-3)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 11 + 12*x + x^2, P(2,x) = 24 + 69*x + 26*x^2 + x^3.","mathematica":"P[1, x_] := 11 + 12*x + x^2;\nP[2, x_] := 24 + 69*x + 26*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + (2*n^2 - 3*n + 4)*x + (n^3 - 2*n^2 + 5*n - 6)*x^2 + (3*n^2 - 4*n + 2)*x^3 + 2*n*x^4)*P[n - 2, x] - (n*x + (n^2 - 2*n + 2)*x^2 - (n^2 - 4*n + 4)*x^3 - (3*n - 4)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n - 2 + (2*n^2 - 3*n + 7)*x + (5*n - 4)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 1)*x - (n - 3)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 1 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 11 + 12*x + x^2; v[2] = 24 + 69*x + 26*x^2 + x^3; for(n = 3, N, my(i = n); v[i] = (-(2*n + (2*n^2 - 3*n + 4)*x + (n^3 - 2*n^2 + 5*n - 6)*x^2 + (3*n^2 - 4*n + 2)*x^3 + 2*n*x^4)*v[i - 2] - (n*x + (n^2 - 2*n + 2)*x^2 - (n^2 - 4*n + 4)*x^3 - (3*n - 4)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n - 2 + (2*n^2 - 3*n + 7)*x + (5*n - 4)*x^2 + 2*x^3)*v[i - 1] + ((n - 1)*x - (n - 3)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 1 + 2*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6a8ffbe72ba42678","steps":["D+1"],"edges":["e-cf9a48dde9435ec6"],"text":"A166341 \u2192 D+1"},"headline":"A166341 \u2192 D+1","chain":null}