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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":"acc4ed1d15cf1c9add5c9a691f75a79f9588e90f14c74082576061257770874f","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-688f70ea6f82c256","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 8).","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:            88,             26,              1;\n  1:           960,            544,             51,              1;\n  2:         11776,          10880,           1840,             84,             1;\n  3:        159744,         219136,          58560,           4640,           125,            1;\n  4:       2326528,        4530176,        1770496,         215040,          9800,          174,           1;\n  5:      35389440,       96600064,       52555776,        9117696,        622720,        18368,         231,         1;\n  6:     551550976,     2120744960,     1556807680,      368492544,      35245056,      1532160,       31584,       296,        1;\n  7:    8707375104,    47706013696,    46387691520,    14514257920,    1860956160,    111562752,     3346560,     50880,      369,      1;\n  8:  138378477568,  1093639602176,  1395147145216,   564375060480,   94085939200,   7453974528,   304668672,   6673920,    77880,    450,   1;\n  9: 2206539448320, 25424595779584, 42391176216576, 21830530236416, 4627748618240, 470338502656, 25118134272, 742612992, 12386880, 114400, 539, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(128-48*x-7*x^2)*P(n-2,x) - (64*x-56*x^2)*d/dx P(n-2,x) + (24-6*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 88 + 26*x + x^2, P(1,x) = 960 + 544*x + 51*x^2 + x^3.","mathematica":"P[0, x_] := 88 + 26*x + x^2;\nP[1, x_] := 960 + 544*x + 51*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[-(128 - 48*x - 7*x^2)*P[n - 2, x] - (64*x - 56*x^2)*D[P[n - 2, x], x] + (24 - 6*x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 88 + 26*x + x^2; v[2] = 960 + 544*x + 51*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(128 - 48*x - 7*x^2)*v[i - 2] - (64*x - 56*x^2)*deriv(v[i - 2], x) + (24 - 6*x)*v[i - 1] + 8*x*deriv(v[i - 1], 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-688f70ea6f82c256","steps":["D+1"],"edges":["e-5aad34cbfa5910da"],"text":"A075503 \u2192 D+1"},"headline":"A075503 \u2192 D+1","chain":null}