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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":"07cbddc73d766a1f56a98fb70babf9745f407e056d52b9890865ed03e7b672b4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e646c30d067149e7","role":"sequence","name":"Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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,  8;\n  2: 1, 16,   24;\n  3: 1, 24,   96,    64;\n  4: 1, 32,  216,   384,   160;\n  5: 1, 40,  384,  1216,  1280,    384;\n  6: 1, 48,  600,  2816,  5280,   3840,    896;\n  7: 1, 56,  864,  5440, 15360,  19584,  10752,   2048;\n  8: 1, 64, 1176,  9344, 36000,  69120,  65408,  28672,  4608;\n  9: 1, 72, 1536, 14784, 72960, 192384, 272384, 202752, 73728, 10240;\n","recurrence_sentence":"The row polynomials satisfy -(n^2+3*n+2+(4*n^3+10*n^2+10*n+12)*x-(8*n^3+20*n^2+4*n-32)*x^2-(8*n^2-8*n)*x^3)*P(n,x) = -(n^2+3*n+2+(4*n^3+26*n^2+42*n+12)*x+(24*n^3+100*n^2+132*n+32)*x^2+(8*n^2-8*n)*x^3)*P(n-1,x) + ((n+2)*x+(4*n^2+28*n+42)*x^2+(48*n^2+200*n+228)*x^3+(16*n^2+112*n+120)*x^4+(16*n+16)*x^5)*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[(-(n^2 + 3*n + 2 + (4*n^3 + 26*n^2 + 42*n + 12)*x + (24*n^3 + 100*n^2 + 132*n + 32)*x^2 + (8*n^2 - 8*n)*x^3)*P[n - 1, x] + ((n + 2)*x + (4*n^2 + 28*n + 42)*x^2 + (48*n^2 + 200*n + 228)*x^3 + (16*n^2 + 112*n + 120)*x^4 + (16*n + 16)*x^5)*D[P[n - 1, x], x])/(-n^2 - 3*n - 2 - (4*n^3 + 10*n^2 + 10*n + 12)*x + (8*n^3 + 20*n^2 + 4*n - 32)*x^2 + (8*n^2 - 8*n)*x^3)]];\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] = (-(n^2 + 3*n + 2 + (4*n^3 + 26*n^2 + 42*n + 12)*x + (24*n^3 + 100*n^2 + 132*n + 32)*x^2 + (8*n^2 - 8*n)*x^3)*v[i - 1] + ((n + 2)*x + (4*n^2 + 28*n + 42)*x^2 + (48*n^2 + 200*n + 228)*x^3 + (16*n^2 + 112*n + 120)*x^4 + (16*n + 16)*x^5)*deriv(v[i - 1], x))/(-n^2 - 3*n - 2 - (4*n^3 + 10*n^2 + 10*n + 12)*x + (8*n^3 + 20*n^2 + 4*n - 32)*x^2 + (8*n^2 - 8*n)*x^3)); 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-e646c30d067149e7","steps":["\u03b8+1"],"edges":["e-422f06a6bd66498d"],"text":"A249252 \u2192 \u03b8+1"},"headline":"A249252 \u2192 \u03b8+1","chain":null}