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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":"cba3d0a83a20c766db131cae4d9713058c063307f5578becbff7dc2fda98e0a7","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-71831045520c8a2d","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.","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:              1541,               1980,                484;\n  1:             47081,             101706,              65340,               10648;\n  2:           1350537,            4143128,            4475064,             1916640,              234256;\n  3:          37218853,          148559070,          227872040,           164085680,            52707600,             5153632;\n  4:         997633165,         4912888596,         9804898620,         10026369760,          5414827440,          1391480640,           113379904;\n  5:       26202352353,       153635507410,       378292421892,        503318129160,        386015235760,        166776685152,         35714669760,          2494357888;\n  6:      677560263953,      4611614014128,     13519924652080,      22193155417664,      22145997683040,      13587736298752,       4892116097792,        897968839680,        54875873536;\n  7:    17306727747101,    134156932262694,    456549787398672,     892315027037280,    1098561193174368,     876981508248384,     448395297858816,     138376998194688,     22224728782080,      1207269217792;\n  8:   437680096735509,   3807480104362220,  14757262548896340,   33480317742569280,   49077326487050400,   48336692499672192,   32155988635774080,   14092423646991360,   3805367450353920,    543271148006400,    26559922791424;\n  9: 10978025471616985, 105918583409993178, 460705092627828620, 1190419178944304760, 2025559223425441440, 2375342601973239360, 1949579930820111744, 1111678464265332480, 426295815321488640, 102322102553960960, 13147161781754880, 584318301411328;\n","recurrence_sentence":"The row polynomials satisfy (22*n+45+22*x)*P(n,x) = (506*n+1541+(484*n+1980)*x+484*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1541 + 1980*x + 484*x^2.","mathematica":"P[0, x_] := 1541 + 1980*x + 484*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((506*n + 1541 + (484*n + 1980)*x + 484*x^2)*P[n - 1, x])/(22*n + 45 + 22*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1541 + 1980*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((506*n + 1541 + (484*n + 1980)*x + 484*x^2)*v[i - 1])/(22*n + 45 + 22*x)); 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-344e4bfa9933088a","steps":["basis A013609","D+1"],"edges":["e-9947ebaa7f991114","e-bf5d7bf2d270203a"],"text":"A038337 \u2192 basis A013609 \u2192 D+1"},"headline":"A038337 \u2192 basis A013609 \u2192 D+1","chain":null}