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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":"013bb32441d3904bc453f9ab866b65a757eaa6bfeeb64bc66e5046774ad8afd2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2c8bb380cc75972b","role":"sequence","name":"Triangle related to the asymptotic expansion of E(x,m=3,n).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:        1;\n  1:        3,        6;\n  2:       11,       36,        18;\n  3:       50,      210,       180,        40;\n  4:      274,     1350,      1530,       600,       75;\n  5:     1764,     9744,     13230,      7000,     1575,      126;\n  6:    13068,    78792,    121842,     78400,    24150,     3528,     196;\n  7:   109584,   708744,   1211112,    897960,   340200,    68796,    7056,    288;\n  8:  1026576,  7036200,  13026240,  10773000,  4745475,  1190700,  170520,  12960,   405;\n  9: 10628640, 76521456, 151371000, 136677200, 67654125, 19879398, 3557400, 380160, 22275, 550;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-8*n^3+24*n^2-32*n+16+(4*n^3-24*n^2+48*n-32)*x+(6*n^2-24*n+24)*x^2+(4*n-8)*x^3+x^4)*P(n-4,x) + (4*n^3-18*n^2+28*n-15+(12*n^2-36*n+28)*x+(12*n-18)*x^2+4*x^3)*P(n-3,x) - (6*n^2-12*n+7+(12*n-12)*x+6*x^2)*P(n-2,x) + (4*n-2+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 6*x, P(2,x) = 11 + 36*x + 18*x^2, P(3,x) = 50 + 210*x + 180*x^2 + 40*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 6*x;\nP[2, x_] := 11 + 36*x + 18*x^2;\nP[3, x_] := 50 + 210*x + 180*x^2 + 40*x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^3 - 24*n^2 + 48*n - 32)*x + (6*n^2 - 24*n + 24)*x^2 + (4*n - 8)*x^3 + x^4)*P[n - 4, x] + (4*n^3 - 18*n^2 + 28*n - 15 + (12*n^2 - 36*n + 28)*x + (12*n - 18)*x^2 + 4*x^3)*P[n - 3, x] - (6*n^2 - 12*n + 7 + (12*n - 12)*x + 6*x^2)*P[n - 2, x] + (4*n - 2 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 6*x; v[3] = 11 + 36*x + 18*x^2; v[4] = 50 + 210*x + 180*x^2 + 40*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^3 - 24*n^2 + 48*n - 32)*x + (6*n^2 - 24*n + 24)*x^2 + (4*n - 8)*x^3 + x^4)*v[i - 4] + (4*n^3 - 18*n^2 + 28*n - 15 + (12*n^2 - 36*n + 28)*x + (12*n - 18)*x^2 + 4*x^3)*v[i - 3] - (6*n^2 - 12*n + 7 + (12*n - 12)*x + 6*x^2)*v[i - 2] + (4*n - 2 + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-efe23a76953cf69d","steps":["D\u00b2\u2218x (shift +1)","\u03b8+1"],"edges":["e-337faa044bd16e1a","e-8af7fd426d6a8dc3"],"text":"A130534 \u2192 D\u00b2\u2218x (shift +1) \u2192 \u03b8+1"},"headline":"A130534 \u2192 D\u00b2\u2218x (shift +1) \u2192 \u03b8+1","chain":null}