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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":"b8c44149e3ffb42c74bae417365dbd0cba299e069f02c90988d133f6463022a7","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-de4284a2d5d7d0a6","role":"sequence","name":"A Motzkin related triangle.","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: 0, 4;\n  2: 0, 4,  9;\n  3: 0, 0, 27,  16;\n  4: 0, 0, 18,  96,  25;\n  5: 0, 0,  0, 160, 250,   36;\n  6: 0, 0,  0,  80, 750,  540,    49;\n  7: 0, 0,  0,   0, 875, 2520,  1029,    64;\n  8: 0, 0,  0,   0, 350, 5040,  6860,  1792,   81;\n  9: 0, 0,  0,   0,   0, 4536, 20580, 16128, 2916, 100;\n","recurrence_sentence":"The row polynomials satisfy (n^4+6*n^3+13*n^2+12*n+4-(2*n^4+10*n^3+13*n^2+3*n-2)*x+(n^4+2*n^3+n^2)*x^2)*P(n,x) = -(4*n^4+12*n^3+4*n^2-12*n-8-(11*n^4+48*n^3+59*n^2+22*n+4)*x+(10*n^4+42*n^3+46*n^2+6*n)*x^2-(3*n^4+8*n^3+4*n^2+n)*x^3)*P(n-1,x) + ((8*n^3+32*n^2+40*n+16)*x-(18*n^3+80*n^2+98*n+28)*x^2+(12*n^3+42*n^2+38*n+10)*x^3-(2*n^3+6*n^2+4*n+1)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^4 + 12*n^3 + 4*n^2 - 12*n - 8 - (11*n^4 + 48*n^3 + 59*n^2 + 22*n + 4)*x + (10*n^4 + 42*n^3 + 46*n^2 + 6*n)*x^2 - (3*n^4 + 8*n^3 + 4*n^2 + n)*x^3)*P[n - 1, x] + ((8*n^3 + 32*n^2 + 40*n + 16)*x - (18*n^3 + 80*n^2 + 98*n + 28)*x^2 + (12*n^3 + 42*n^2 + 38*n + 10)*x^3 - (2*n^3 + 6*n^2 + 4*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 - (2*n^4 + 10*n^3 + 13*n^2 + 3*n - 2)*x + (n^4 + 2*n^3 + n^2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(4*n^4 + 12*n^3 + 4*n^2 - 12*n - 8 - (11*n^4 + 48*n^3 + 59*n^2 + 22*n + 4)*x + (10*n^4 + 42*n^3 + 46*n^2 + 6*n)*x^2 - (3*n^4 + 8*n^3 + 4*n^2 + n)*x^3)*v[i - 1] + ((8*n^3 + 32*n^2 + 40*n + 16)*x - (18*n^3 + 80*n^2 + 98*n + 28)*x^2 + (12*n^3 + 42*n^2 + 38*n + 10)*x^3 - (2*n^3 + 6*n^2 + 4*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 - (2*n^4 + 10*n^3 + 13*n^2 + 3*n - 2)*x + (n^4 + 2*n^3 + n^2)*x^2)); 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-de4284a2d5d7d0a6","steps":["(\u03b8+1)\u00b2"],"edges":["e-a9d09ef5df43e2e6"],"text":"A107131 \u2192 (\u03b8+1)\u00b2"},"headline":"A107131 \u2192 (\u03b8+1)\u00b2","chain":null}