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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":"4e163bbdb8ba04ff38c609f7523d4a6a252421d21bf4565029b6adfa7dbaf80a","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-c63944481deaae45","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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:    2,     2;\n   3:    6,    10,      3;\n   4:   16,    40,     24,      4;\n   5:   44,   144,    129,     44,      5;\n   6:  120,   488,    582,    300,     70,     6;\n   7:  328,  1592,   2370,   1636,    580,   102,     7;\n   8:  896,  5056,   9024,   7808,   3720,   996,   140,    8;\n   9: 2448, 15744,  32760,  34080,  20405,  7356,  1575,  184,   9;\n  10: 6688, 48288, 114768, 139488, 100610, 45654, 13174, 2344, 234, 10;\n","recurrence_sentence":"The row polynomials satisfy (48*n-48+(12*n^2+8*n-20)*x+(n^3+2*n^2-3*n)*x^2)*P(n,x) = -(96*n-288+(36*n^2-92*n-144)*x+(2*n^3-54*n-16)*x^2-(2*n^2+6*n)*x^3)*P(n-1,x) + (288+(144*n+288)*x+(12*n^2+132*n+88)*x^2+(8*n^2+36*n+8)*x^3+(n^2+3*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(96*n - 288 + (36*n^2 - 92*n - 144)*x + (2*n^3 - 54*n - 16)*x^2 - (2*n^2 + 6*n)*x^3)*P[n - 1, x] + (288 + (144*n + 288)*x + (12*n^2 + 132*n + 88)*x^2 + (8*n^2 + 36*n + 8)*x^3 + (n^2 + 3*n)*x^4)*D[P[n - 1, x], x])/(48*n - 48 + (12*n^2 + 8*n - 20)*x + (n^3 + 2*n^2 - 3*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(96*n - 288 + (36*n^2 - 92*n - 144)*x + (2*n^3 - 54*n - 16)*x^2 - (2*n^2 + 6*n)*x^3)*v[i - 1] + (288 + (144*n + 288)*x + (12*n^2 + 132*n + 88)*x^2 + (8*n^2 + 36*n + 8)*x^3 + (n^2 + 3*n)*x^4)*deriv(v[i - 1], x))/(48*n - 48 + (12*n^2 + 8*n - 20)*x + (n^3 + 2*n^2 - 3*n)*x^2)); 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-c72653d9ab0ee8be","steps":["basis A007318","\u03b8+1"],"edges":["e-3a20d12bade77e7f","e-4b2630e19acb34db"],"text":"A063967 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A063967 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}