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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":"d3ee411898fb2182d4633c6029915522c4d43ecd1d4ec9b528eaac8115ff2681","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-313c8ccf28f71545","role":"sequence","name":"Triangle read by rows, Lah numbers with level 2.","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: 0,            2;\n   3: 0,            8,             3;\n   4: 0,           72,            60,             4;\n   5: 0,         1152,          1608,           224,            5;\n   6: 0,        28800,         59616,         13120,          600,          6;\n   7: 0,      1036800,       2964384,        919168,        65000,       1320,         7;\n   8: 0,     50803200,     191275776,      78405120,      7648960,     237720,      2548,       8;\n   9: 0,   3251404800,   15569542656,    8095546368,   1023530560,   43410432,    707952,    4480,    9;\n  10: 0, 263363788800, 1561831372800, 1000320500736, 157507833600, 8564599680, 189404096, 1817088, 7344, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-14*n^3+73*n^2-168*n+144+(2*n^2-16*n+32)*x+x^2)*P(n-2,x) - ((2*n^3-21*n^2+72*n-80)*x+(2*n-6)*x^2)*d/dx P(n-2,x) - ((n^2-8*n+16)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2-12*n+20+2*x)*P(n-1,x) + (2*n-3)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 14*n^3 + 73*n^2 - 168*n + 144 + (2*n^2 - 16*n + 32)*x + x^2)*P[n - 2, x] - ((2*n^3 - 21*n^2 + 72*n - 80)*x + (2*n - 6)*x^2)*D[P[n - 2, x], x] - ((n^2 - 8*n + 16)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 - 12*n + 20 + 2*x)*P[n - 1, x] + (2*n - 3)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = -(n^4 - 14*n^3 + 73*n^2 - 168*n + 144 + (2*n^2 - 16*n + 32)*x + x^2)*v[i - 2] - ((2*n^3 - 21*n^2 + 72*n - 80)*x + (2*n - 6)*x^2)*deriv(v[i - 2], x) - ((n^2 - 8*n + 16)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 12*n + 20 + 2*x)*v[i - 1] + (2*n - 3)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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-313c8ccf28f71545","steps":["\u03b8+1"],"edges":["e-164e954b4a5d0506"],"text":"A390433 \u2192 \u03b8+1"},"headline":"A390433 \u2192 \u03b8+1","chain":null}