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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"d2e6a6f7cc14da081595c05aa18c72fa3bebb1e1f6fd309d996c3c1c3a489e7b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f39c1a4c1fe3654f","role":"sequence","name":"derivative 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:      1;\n   3:      3,       4;\n   4:      8,      24,       9;\n   5:     24,     116,      90,      16;\n   6:     89,     580,     675,     240,      25;\n   7:    415,    3256,    4905,    2560,     525,     36;\n   8:   2372,   20972,   37611,   25200,    7525,   1008,    49;\n   9:  16072,  154472,  313740,  250544,   95900,  18648,  1764,   64;\n  10: 125673, 1286760, 2870334, 2611056, 1194375, 297864, 40866, 2880, 81;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^3-15*n^2+74*n-120)*P(n-6,x) + (3*n^3-39*n^2+168*n-240+(3*n^2-27*n+60)*x)*P(n-5,x) - (3*n^3-30*n^2+97*n-100+(6*n^2-45*n+84)*x+(3*n-12)*x^2)*P(n-4,x) + (n^3-3*n^2-12*n+37+(3*n^2-12*n+6)*x+(3*n-9)*x^2+x^3)*P(n-3,x) - (3*n^2-12*n+10+(6*n-15)*x+3*x^2)*P(n-2,x) + (3*n-6+3*x)*P(n-1,x) for n >= 7, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 3 + 4*x, P(4,x) = 8 + 24*x + 9*x^2, P(5,x) = 24 + 116*x + 90*x^2 + 16*x^3, P(6,x) = 89 + 580*x + 675*x^2 + 240*x^3 + 25*x^4.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[3, x_] := 3 + 4*x;\nP[4, x_] := 8 + 24*x + 9*x^2;\nP[5, x_] := 24 + 116*x + 90*x^2 + 16*x^3;\nP[6, x_] := 89 + 580*x + 675*x^2 + 240*x^3 + 25*x^4;\nP[n_, x_] := P[n, x] = Expand[-(n^3 - 15*n^2 + 74*n - 120)*P[n - 6, x] + (3*n^3 - 39*n^2 + 168*n - 240 + (3*n^2 - 27*n + 60)*x)*P[n - 5, x] - (3*n^3 - 30*n^2 + 97*n - 100 + (6*n^2 - 45*n + 84)*x + (3*n - 12)*x^2)*P[n - 4, x] + (n^3 - 3*n^2 - 12*n + 37 + (3*n^2 - 12*n + 6)*x + (3*n - 9)*x^2 + x^3)*P[n - 3, x] - (3*n^2 - 12*n + 10 + (6*n - 15)*x + 3*x^2)*P[n - 2, x] + (3*n - 6 + 3*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] = 1; v[3] = 3 + 4*x; v[4] = 8 + 24*x + 9*x^2; v[5] = 24 + 116*x + 90*x^2 + 16*x^3; v[6] = 89 + 580*x + 675*x^2 + 240*x^3 + 25*x^4; for(n = 7, N, my(i = n); v[i] = -(n^3 - 15*n^2 + 74*n - 120)*v[i - 6] + (3*n^3 - 39*n^2 + 168*n - 240 + (3*n^2 - 27*n + 60)*x)*v[i - 5] - (3*n^3 - 30*n^2 + 97*n - 100 + (6*n^2 - 45*n + 84)*x + (3*n - 12)*x^2)*v[i - 4] + (n^3 - 3*n^2 - 12*n + 37 + (3*n^2 - 12*n + 6)*x + (3*n - 9)*x^2 + x^3)*v[i - 3] - (3*n^2 - 12*n + 10 + (6*n - 15)*x + 3*x^2)*v[i - 2] + (3*n - 6 + 3*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":["D","\u03b8+1"],"edges":["e-d161d3edb4630d0d","e-c7162ac69f8e946c"],"text":"A094816 \u2192 D \u2192 \u03b8+1"},"headline":"A094816 \u2192 D \u2192 \u03b8+1","chain":null}