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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":"f31d699b0fa2961e69994c6c3348a2e2e7bf543649ef9bd2c8bbc4855d97ad20","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-7196be931c2ed996","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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,  6;\n  2: 1,  8,   9;\n  3: 1, 14,  45,   36;\n  4: 1, 16,  66,   96,    45;\n  5: 1, 22, 138,  360,   405,   162;\n  6: 1, 24, 171,  544,   855,   648,   189;\n  7: 1, 30, 279, 1228,  2895,  3726,  2457,   648;\n  8: 1, 32, 324, 1600,  4430,  7200,  6804,  3456,   729;\n  9: 1, 38, 468, 2896, 10430, 23148, 32004, 26784, 12393, 2430;\n","recurrence_sentence":"The row polynomials satisfy (4+(12*n+13)*x+(8*n^2+38*n)*x^2+(24*n^2+18*n-21)*x^3+(18*n^2-18*n)*x^4)*P(n,x) = (4+(12*n+41)*x+(8*n^2+102*n+158)*x^2+(56*n^2+318*n+280)*x^3+(138*n^2+444*n+222)*x^4+(144*n^2+270*n+63)*x^5+(54*n^2+54*n)*x^6)*P(n-2,x) + (4*x+22*x^2+36*x^3+18*x^4)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4 + (12*n + 41)*x + (8*n^2 + 102*n + 158)*x^2 + (56*n^2 + 318*n + 280)*x^3 + (138*n^2 + 444*n + 222)*x^4 + (144*n^2 + 270*n + 63)*x^5 + (54*n^2 + 54*n)*x^6)*P[n - 2, x] + (4*x + 22*x^2 + 36*x^3 + 18*x^4)*P[n - 1, x])/(4 + (12*n + 13)*x + (8*n^2 + 38*n)*x^2 + (24*n^2 + 18*n - 21)*x^3 + (18*n^2 - 18*n)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((4 + (12*n + 41)*x + (8*n^2 + 102*n + 158)*x^2 + (56*n^2 + 318*n + 280)*x^3 + (138*n^2 + 444*n + 222)*x^4 + (144*n^2 + 270*n + 63)*x^5 + (54*n^2 + 54*n)*x^6)*v[i - 2] + (4*x + 22*x^2 + 36*x^3 + 18*x^4)*v[i - 1])/(4 + (12*n + 13)*x + (8*n^2 + 38*n)*x^2 + (24*n^2 + 18*n - 21)*x^3 + (18*n^2 - 18*n)*x^4)); 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-7196be931c2ed996","steps":["\u03b8+1"],"edges":["e-565e2f08268e5565"],"text":"A152842 \u2192 \u03b8+1"},"headline":"A152842 \u2192 \u03b8+1","chain":null}