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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":"671e36616d4702ce744909baccbfec597ef7fe0ed4de0ba22c2ba01a751f9847","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-1a01d480c8d600c3","role":"sequence","name":"basis_map_A013611 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 1","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:          5,           4;\n  2:         24,          52,          16;\n  3:        184,         540,         384,          64;\n  4:       1952,        6760,        6768,        2432,         256;\n  5:      26496,      102376,      125200,       64320,       14080,       1024;\n  6:     438496,     1835424,     2571744,     1643840,      518400,      76800,       4096;\n  7:    8564384,    38148064,    59115168,    43978816,    17454080,    3741696,     401408,     16384;\n  8:  192842688,   903346240,  1514506432,  1262354688,   589293824,  160219136,   24944640,   2031616,    65536;\n  9: 4918444480, 24029020224, 42945741504, 39146336256, 20661758208, 6658089984, 1322065920, 156598272, 10027008, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(24*n-40)*x+16*x^2)*P(n-2,x) + (6*n-7+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (24*n - 40)*x + 16*x^2)*P[n - 2, x] + (6*n - 7 + 8*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (24*n - 40)*x + 16*x^2)*v[i - 2] + (6*n - 7 + 8*x)*v[i - 1]); 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-1a0f876fa1441737","steps":["basis A013611","D+1"],"edges":["e-422c7a2dd7cada12","e-d76fed48a50545f9"],"text":"basis-map coefficients of A371080, shifted \u2192 basis A013611 \u2192 D+1"},"headline":"basis-map coefficients of A371080, shifted \u2192 basis A013611 \u2192 D+1","chain":null}