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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":"5bab5ab328fe573949172e29a014d06446ed1d5b2bd18073d3671c900f8efcab","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-344e4bfa9933088a","role":"sequence","name":"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:           408,            506,             121;\n  1:          6480,          13464,            8349,            1331;\n  2:         96768,         285120,          296208,          122452,            14641;\n  3:       1389312,        5322240,         7840800,         5430480,          1683715,           161051;\n  4:      19408896,       91694592,       175633920,       172497600,         89602920,         22225038,          1771561;\n  5:     265752576,     1494484992,      3530241792,      4507937280,       3320578800,       1379884968,        285221321,        19487171;\n  6:    3583180800,    23386226688,     65757339648,    103553759232,      99174620160,      58442186880,      20238312864,      3585639464,       214358881;\n  7:   47727968256,   354734899200,   1157618221056,   2169992208384,    2562955540992,    1963657479168,     964296083520,    286227567648,     44372288367,     2357947691;\n  8:  629493202944,  5250076508160,  19510419456000,  42446001438720,   59674785730560,   56385021901824,   36000387118080,  15153224169600,   3935629055160,   542327968930,   25937424601;\n  9: 8235009441792, 76168677556224, 317629628743680, 786920251392000, 1283991543521280, 1444129814679552, 1137097941686784, 622292405898240, 229192515565200, 52912346186040, 6562168424053, 285311670611;\n","recurrence_sentence":"The row polynomials satisfy (11*n+23+11*x)*P(n,x) = (132*n+408+(121*n+506)*x+121*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 408 + 506*x + 121*x^2.","mathematica":"P[0, x_] := 408 + 506*x + 121*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((132*n + 408 + (121*n + 506)*x + 121*x^2)*P[n - 1, x])/(11*n + 23 + 11*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 408 + 506*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((132*n + 408 + (121*n + 506)*x + 121*x^2)*v[i - 1])/(11*n + 23 + 11*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":["D+1"],"edges":["e-991f2aa4fae51627"],"text":"A038337 \u2192 D+1"},"headline":"A038337 \u2192 D+1","chain":null}