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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":"cc9b5fe0d3a8fb11dfbfc305d881fbdf043876d817da504eead5f5830909aff9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-318700a428d7b97b","role":"sequence","name":"Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","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:  3,    4;\n   3:  5,   20,     9;\n   4:  7,   56,    63,    16;\n   5:  9,  120,   243,   144,    25;\n   6: 11,  220,   693,   704,   275,    36;\n   7: 13,  364,  1638,  2496,  1625,   468,    49;\n   8: 15,  560,  3402,  7200,  6875,  3240,   735,   64;\n   9: 17,  816,  6426, 17952, 23375, 15912,  5831, 1088,   81;\n  10: 19, 1140, 11286, 40128, 67925, 62244, 32585, 9728, 1539, 100;\n","recurrence_sentence":"The row polynomials satisfy (8*n-12-(20*n^3-90*n^2+134*n-66)*x+(2*n^5-15*n^4+44*n^3-63*n^2+44*n-12)*x^2)*P(n,x) = (8*n-4-(20*n^3-70*n^2+58*n-6)*x+(2*n^5-13*n^4+14*n^3+17*n^2-24*n+4)*x^2+(n^5-5*n^4+8*n^3-4*n^2)*x^3)*P(n-1,x) + (16*x-(40*n^2-80*n+12)*x^2+(4*n^4-16*n^3+6*n^2+20*n-4)*x^3+(n^4-4*n^3+4*n^2)*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[((8*n - 4 - (20*n^3 - 70*n^2 + 58*n - 6)*x + (2*n^5 - 13*n^4 + 14*n^3 + 17*n^2 - 24*n + 4)*x^2 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^3)*P[n - 1, x] + (16*x - (40*n^2 - 80*n + 12)*x^2 + (4*n^4 - 16*n^3 + 6*n^2 + 20*n - 4)*x^3 + (n^4 - 4*n^3 + 4*n^2)*x^4)*D[P[n - 1, x], x])/(8*n - 12 - (20*n^3 - 90*n^2 + 134*n - 66)*x + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*x^2)]];\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] = ((8*n - 4 - (20*n^3 - 70*n^2 + 58*n - 6)*x + (2*n^5 - 13*n^4 + 14*n^3 + 17*n^2 - 24*n + 4)*x^2 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^3)*v[i - 1] + (16*x - (40*n^2 - 80*n + 12)*x^2 + (4*n^4 - 16*n^3 + 6*n^2 + 20*n - 4)*x^3 + (n^4 - 4*n^3 + 4*n^2)*x^4)*deriv(v[i - 1], x))/(8*n - 12 - (20*n^3 - 90*n^2 + 134*n - 66)*x + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*x^2)); 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-318700a428d7b97b","steps":["(\u03b8+1)\u00b2"],"edges":["e-c38cb060e3b23574"],"text":"A111125 \u2192 (\u03b8+1)\u00b2"},"headline":"A111125 \u2192 (\u03b8+1)\u00b2","chain":null}