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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":"5ae38a21806ddc764079caa13d0e075b47e013b5ed709e6c9ae1c361ed0800ea","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-c21cdce554fb6206","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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:             901,             1260,               324;\n  1:           20519,            48654,             34020,              5832;\n  2:          437257,          1477368,           1751544,            816480,             104976;\n  3:         8936747,         39353130,          66481560,          52546320,           18370800,            1889568;\n  4:       177482125,        965168676,        2125069020,        2393336160,         1418750640,          396809280,           34012224;\n  5:      3451672367,      22362747750,       60805626588,       89252898840,        75390089040,        35752516128,         8332994880,         612220032;\n  6:     66064526353,     497040820848,     1610117838000,     2918670076224,      3213104358240,      2171234564352,       858060387072,      171421608960,       11019960576;\n  7:   1248660581939,   10702453269186,    40260306488688,    86946363252000,    118206138087072,    104104581206976,     58623333237504,    19857968957952,     3471287581440,     198359290368;\n  8:  23361811669909,  224758904749020,   963220794226740,  2415618389321280,   3912586346340000,   4255420971134592,   3123137436209280,  1507457140392960,   446804301553920,   69425751628800,    3570467226624;\n  9: 433438688596535, 4625638710641982, 22251131570152980, 63572572418964840, 119573110271403360, 154938419315064000, 140428892047441536, 88340173195633920, 37309564224725760, 9829694634186240, 1374629882250240, 64268410079232;\n","recurrence_sentence":"The row polynomials satisfy (18*n+35+18*x)*P(n,x) = (306*n+901+(324*n+1260)*x+324*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 901 + 1260*x + 324*x^2.","mathematica":"P[0, x_] := 901 + 1260*x + 324*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((306*n + 901 + (324*n + 1260)*x + 324*x^2)*P[n - 1, x])/(18*n + 35 + 18*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 901 + 1260*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((306*n + 901 + (324*n + 1260)*x + 324*x^2)*v[i - 1])/(18*n + 35 + 18*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-1f2dec04776afb2b","steps":["basis A013610","D+1"],"edges":["e-572c9115d911d6cf","e-434c5fe053ebc993"],"text":"A038320 \u2192 basis A013610 \u2192 D+1"},"headline":"A038320 \u2192 basis A013610 \u2192 D+1","chain":null}