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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":"7e4adbe5afba07ddbf14b1bbd1f384eeff39126de65ae6cff28f74ef9c6a02f7","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-b11b90d367f09b77","role":"sequence","name":"basis_map_A013609 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:       0;\n   2:       2;\n   3:      10,        8;\n   4:      46,       72,       24;\n   5:     232,      520,      336,       64;\n   6:    1338,     3760,     3480,     1280,      160;\n   7:    8858,    29072,    33960,    17920,     4320,     384;\n   8:   66750,   245448,   338016,   226240,    78400,   13440,     896;\n   9:  566144,  2272528,  3540768,  2812096,  1240960,  306432,   39424,   2048;\n  10: 5346642, 23026272, 39521136, 35753088, 18812640, 5935104, 1102080, 110592, 4608;\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+(4*n-12)*x)*P(n-3,x) - (n^2-4+(4*n-4)*x+4*x^2)*P(n-2,x) + (2*n-1+4*x)*P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 2, P(3,x) = 10 + 8*x, P(4,x) = 46 + 72*x + 24*x^2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\nP[3, x_] := 10 + 8*x;\nP[4, x_] := 46 + 72*x + 24*x^2;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (4*n - 12)*x)*P[n - 3, x] - (n^2 - 4 + (4*n - 4)*x + 4*x^2)*P[n - 2, x] + (2*n - 1 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; v[3] = 10 + 8*x; v[4] = 46 + 72*x + 24*x^2; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (4*n - 12)*x)*v[i - 3] - (n^2 - 4 + (4*n - 4)*x + 4*x^2)*v[i - 2] + (2*n - 1 + 4*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 A013609","D"],"edges":["e-9a46600ceb40809a","e-467eaade26420dde"],"text":"A094816 \u2192 basis A013609 \u2192 D"},"headline":"A094816 \u2192 basis A013609 \u2192 D","chain":null}