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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":"44bdfa2a3ad43ffd7be9306fe6b083d69694abf76c6005237780bb1631af8045","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:       1;\n   2:       4,        2;\n   3:      15,       18,        4;\n   4:      62,      118,       60,        8;\n   5:     297,      752,      596,      176,       16;\n   6:    1664,     5098,     5360,     2440,      480,      32;\n   7:   10815,    37930,    48496,    29240,     8800,    1248,      64;\n   8:   80450,   312198,   460740,   338912,   134960,   29120,    3136,    128;\n   9:  675745,  2838672,  4677032,  3992352,  1943984,  554624,   90496,   7680,   256;\n  10: 6333052, 28372914, 51034272, 48926800, 27750912, 9697632, 2091264, 268032, 18432, 512;\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) = 1, P(2,x) = 4 + 2*x, P(3,x) = 15 + 18*x + 4*x^2, P(4,x) = 62 + 118*x + 60*x^2 + 8*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 2*x;\nP[3, x_] := 15 + 18*x + 4*x^2;\nP[4, x_] := 62 + 118*x + 60*x^2 + 8*x^3;\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] = 1; v[2] = 4 + 2*x; v[3] = 15 + 18*x + 4*x^2; v[4] = 62 + 118*x + 60*x^2 + 8*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 + (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+1"],"edges":["e-9a46600ceb40809a","e-01c7f7a6f74e9e62"],"text":"A094816 \u2192 basis A013609 \u2192 D+1"},"headline":"A094816 \u2192 basis A013609 \u2192 D+1","chain":null}