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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":"0cccffd2ef79c6e705b929d8b3b2e66c049afb56b7a96a22f93a703a62a847ca","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-2f0c73de4c291a40","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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:        11,           8;\n  2:        80,         260,           96;\n  3:       592,        3668,         6176,          1536;\n  4:      4752,       44488,       141776,        163456,         30720;\n  5:     41904,      530872,      2541296,       5520640,       4875264,         737280;\n  6:    405792,     6523040,     42075616,     136746560,     226868224,      163049472,       20643840;\n  7:   4297248,    84032928,    685616352,    2993212480,    7407657984,     9981194240,     6063980544,      660602880;\n  8:  49504320,  1143199296,  11319857088,   62362206976,  206834168064,   414822379520,   471794171904,   248691032064,    23781703680;\n  9: 617123520, 16462961856, 192156587328, 1282406592256, 5357379189504, 14373543229440, 24291925213184, 23944864333824, 11160508170240, 951268147200;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(8*n^3+16*n^2+8*n)*x+(16*n^3+16*n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(8*n^2+8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 11 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 11 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (8*n^3 + 16*n^2 + 8*n)*x + (16*n^3 + 16*n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (8*n^2 + 8*n)*x)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 11 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (8*n^3 + 16*n^2 + 8*n)*x + (16*n^3 + 16*n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (8*n^2 + 8*n)*x)*v[i - 1])/(n)); 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-2feb6dc01f424b0a","steps":["basis A013611","D+1"],"edges":["e-39cad9e518a532bb","e-dfad10853dccf1ba"],"text":"A145324 \u2192 basis A013611 \u2192 D+1"},"headline":"A145324 \u2192 basis A013611 \u2192 D+1","chain":null}