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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":"61c6b50e0c2503ee49eb082e8a7ba6ad00c8c045358c18ff78017a40146c75c1","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-8cf5a16a6dfee934","role":"sequence","name":"Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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: 1,     4;\n  2: 1,    16,      18;\n  3: 1,    52,     162,       96;\n  4: 1,   160,    1044,     1536,       600;\n  5: 1,   484,    5940,    16320,     15000,      4320;\n  6: 1,  1456,   31878,   145920,    237000,    155520,      35280;\n  7: 1,  4372,  165942,  1191456,   3045000,   3417120,    1728720,    322560;\n  8: 1, 13120,  849384,  9225216,  34851600,  59512320,   50379840,  20643840,   3265920;\n  9: 1, 39364, 4305960, 69106560, 371322000, 905567040, 1140955200, 770273280, 264539520, 36288000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+4*x-2*x^2)*P(n-2,x) + (2*x-x^2-x^3)*d/dx P(n-2,x) + 4*x*P(n-1,x) + (2*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 4*x - 2*x^2)*P[n - 2, x] + (2*x - x^2 - x^3)*D[P[n - 2, x], x] + 4*x*P[n - 1, x] + (2*x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x - 2*x^2)*v[i - 2] + (2*x - x^2 - x^3)*deriv(v[i - 2], x) + 4*x*v[i - 1] + (2*x + x^2)*deriv(v[i - 1], x)); 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-8cf5a16a6dfee934","steps":["(\u03b8+1)\u00b2"],"edges":["e-0b9f50e4a34af92e"],"text":"A225476 \u2192 (\u03b8+1)\u00b2"},"headline":"A225476 \u2192 (\u03b8+1)\u00b2","chain":null}