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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":"f4b396ef9bb88cfd877af13879d8b58e43731533f26ee635ff4bbe0137e83508","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-ee693a689121a12c","role":"sequence","name":"Triangle of numbers used for basis change between certain falling factorials.","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: 1,   8;\n   3: 1,  24,    90;\n   4: 1,  48,   504,    1344;\n   5: 1,  80,  1620,   11520,    25200;\n   6: 1, 120,  3960,   52800,   297000,     570240;\n   7: 1, 168,  8190,  174720,  1801800,    8648640,    15135120;\n   8: 1, 224, 15120,  470400,  7644000,   66044160,   282522240,    461260800;\n   9: 1, 288, 25704, 1096704, 25704000,  345461760,  2619751680,  10265149440,  15878903040;\n  10: 1, 360, 41040, 2298240, 73256400, 1406522880, 16409433600, 112521830400, 411407942400, 609493248000;\n","recurrence_sentence":"The row polynomials satisfy (1+(2*n^2-3*n+1)*x)*P(n,x) = (1+(2*n^2+5*n-7)*x+(8*n^3-7*n^2-11*n+10)*x^2+(2*n^4-7*n^3+6*n^2+n-2)*x^3)*P(n-1,x) - (x^2+(2*n^2+2*n-3)*x^3+(2*n^3-3*n^2+1)*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[((1 + (2*n^2 + 5*n - 7)*x + (8*n^3 - 7*n^2 - 11*n + 10)*x^2 + (2*n^4 - 7*n^3 + 6*n^2 + n - 2)*x^3)*P[n - 1, x] - (x^2 + (2*n^2 + 2*n - 3)*x^3 + (2*n^3 - 3*n^2 + 1)*x^4)*D[P[n - 1, x], x])/(1 + (2*n^2 - 3*n + 1)*x)]];\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] = ((1 + (2*n^2 + 5*n - 7)*x + (8*n^3 - 7*n^2 - 11*n + 10)*x^2 + (2*n^4 - 7*n^3 + 6*n^2 + n - 2)*x^3)*v[i - 1] - (x^2 + (2*n^2 + 2*n - 3)*x^3 + (2*n^3 - 3*n^2 + 1)*x^4)*deriv(v[i - 1], x))/(1 + (2*n^2 - 3*n + 1)*x)); 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-ee693a689121a12c","steps":["\u03b8+1"],"edges":["e-fa7eed07e103faab"],"text":"A089503 \u2192 \u03b8+1"},"headline":"A089503 \u2192 \u03b8+1","chain":null}