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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":"ada2ca59fb85f9993df0ee05251eda0e0bab6bc3859aefa6d64b907768da47e7","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-3deb3ea1ada5b4a1","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+4x)^n.","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:       21,         16;\n  2:      185,        672,         256;\n  3:     1325,       8880,       16128,         4096;\n  4:     8625,      84800,      284160,       344064,        65536;\n  5:    53125,     690000,     3392000,      7577600,      6881280,       1048576;\n  6:   315625,    5100000,    33120000,    108544000,    181862400,     132120576,      16777216;\n  7:  1828125,   35350000,   285600000,   1236480000,   3039232000,    4073717760,    2466250752,     268435456;\n  8: 10390625,  234000000,  2262400000,  12185600000,  39567360000,   77804339200,   86905978880,   45097156608,   4294967296;\n  9: 58203125, 1496250000, 16848000000, 108595200000, 438681600000, 1139539968000, 1867304140800, 1787780136960, 811748818944, 68719476736;\n","recurrence_sentence":"The row polynomials satisfy (16*n-11+16*x)*P(n,x) = (80*n+25+(256*n+160)*x+256*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((80*n + 25 + (256*n + 160)*x + 256*x^2)*P[n - 1, x])/(16*n - 11 + 16*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((80*n + 25 + (256*n + 160)*x + 256*x^2)*v[i - 1])/(16*n - 11 + 16*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-2e0409c0014ec240","steps":["basis A013611","D+1"],"edges":["e-189a8da608edbb49","e-1f5e57e0622fb8aa"],"text":"A013611 \u2192 basis A013611 \u2192 D+1"},"headline":"A013611 \u2192 basis A013611 \u2192 D+1","chain":null}