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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":"c8f0d92cfd95f1b6b7120611d15d9e9ac35e751c774cacc27441a1ea7d70f9ff","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-735a959cc5ec274b","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        1;\n   2:        6,         4;\n   3:       25,        52,        16;\n   4:      108,       380,       336,        64;\n   5:      529,      2544,      3728,      1920,       256;\n   6:     3002,     17716,     35360,     29760,     10240,      1024;\n   7:    19673,    134004,    329824,    377280,    209920,     52224,      4096;\n   8:   147200,   1115292,   3195024,   4521216,   3413760,   1361920,    258048,    16384;\n   9:  1241889,  10222400,  32871200,  54435584,  50959104,  27553792,   8314880,  1245184,   65536;\n  10: 11679694, 102798372, 362221632, 677439104, 745016832, 500247552, 204374016, 48463872, 5898240, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(8*n-24)*x)*P(n-3,x) - (n^2-4+(8*n-8)*x+16*x^2)*P(n-2,x) + (2*n-1+8*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 6 + 4*x, P(3,x) = 25 + 52*x + 16*x^2, P(4,x) = 108 + 380*x + 336*x^2 + 64*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 6 + 4*x;\nP[3, x_] := 25 + 52*x + 16*x^2;\nP[4, x_] := 108 + 380*x + 336*x^2 + 64*x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (8*n - 24)*x)*P[n - 3, x] - (n^2 - 4 + (8*n - 8)*x + 16*x^2)*P[n - 2, x] + (2*n - 1 + 8*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 4*x; v[3] = 25 + 52*x + 16*x^2; v[4] = 108 + 380*x + 336*x^2 + 64*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (8*n - 24)*x)*v[i - 3] - (n^2 - 4 + (8*n - 8)*x + 16*x^2)*v[i - 2] + (2*n - 1 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3272d5bc697b9cb4","steps":["basis A013611","D+1"],"edges":["e-91983714a64825b5","e-1aca8382864ed224"],"text":"A094816 \u2192 basis A013611 \u2192 D+1"},"headline":"A094816 \u2192 basis A013611 \u2192 D+1","chain":null}