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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":"a1ba82c1772722eb40d247e2ad04190504f1f7c089bbe8bb08c6c41bd0e4fb13","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-67a03de73dfc9066","role":"sequence","name":"one_plus_derivative of Number triangle T(n,k)=C(n,k)C(2,n-k).","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,  2;\n  2: 5, 12,  3;\n  3: 3, 30, 27,   4;\n  4: 0, 24, 90,  48,   5;\n  5: 0,  0, 90, 200,  75,    6;\n  6: 0,  0,  0, 240, 375,  108,    7;\n  7: 0,  0,  0,   0, 525,  630,  147,    8;\n  8: 0,  0,  0,   0,   0, 1008,  980,  192,   9;\n  9: 0,  0,  0,   0,   0,    0, 1764, 1440, 243, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x^5*P(n-5,x) - 5*x^4*P(n-4,x) + 10*x^3*P(n-3,x) - 10*x^2*P(n-2,x) + 5*x*P(n-1,x) for n >= 5, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 5 + 12*x + 3*x^2, P(3,x) = 3 + 30*x + 27*x^2 + 4*x^3, P(4,x) = 24*x + 90*x^2 + 48*x^3 + 5*x^4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[2, x_] := 5 + 12*x + 3*x^2;\nP[3, x_] := 3 + 30*x + 27*x^2 + 4*x^3;\nP[4, x_] := 24*x + 90*x^2 + 48*x^3 + 5*x^4;\nP[n_, x_] := P[n, x] = Expand[x^5*P[n - 5, x] - 5*x^4*P[n - 4, x] + 10*x^3*P[n - 3, x] - 10*x^2*P[n - 2, x] + 5*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 + 2*x; v[3] = 5 + 12*x + 3*x^2; v[4] = 3 + 30*x + 27*x^2 + 4*x^3; v[5] = 24*x + 90*x^2 + 48*x^3 + 5*x^4; for(n = 5, N, my(i = n + 1); v[i] = x^5*v[i - 5] - 5*x^4*v[i - 4] + 10*x^3*v[i - 3] - 10*x^2*v[i - 2] + 5*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-1815d3e1c9c57d5b","steps":["D+1","\u03b8+1"],"edges":["e-18255f76f0c1af14","e-d5f418bb9635ea84"],"text":"A106246 \u2192 D+1 \u2192 \u03b8+1"},"headline":"A106246 \u2192 D+1 \u2192 \u03b8+1","chain":null}