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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":"af70db08d5be432ff3f0402531a44e81f5fe3eaf9fbb9fea5ac0bdbcf304d8c3","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-7f247edfefc78f8c","role":"sequence","name":"basis_map_A013610 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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,    6;\n   3:  2,   18,    27;\n   4:  3,   42,   135,    108;\n   5:  5,   90,   432,    756,    405;\n   6:  8,  180,  1161,   3132,   3645,    1458;\n   7: 13,  348,  2808,  10260,  18630,   16038,    5103;\n   8: 21,  654,  6345,  29268,  72495,   97686,   66339,   17496;\n   9: 34, 1206, 13662,  76140, 239355,  441774,  469476,  262440,   59049;\n  10: 55, 2190, 28377, 185436, 707130, 1662120, 2423925, 2117016, 1003833, 196830;\n","recurrence_sentence":"The row polynomials satisfy (30*n-30+(45*n^2+27*n-72)*x+(27*n^3+27*n^2-54*n)*x^2)*P(n,x) = -(60*n-150+(135*n^2-261*n-360)*x+(54*n^3-27*n^2-675*n-162)*x^2-(162*n^2+324*n)*x^3)*P(n-1,x) + (25+(75*n+135)*x+(45*n^2+360*n+207)*x^2+(162*n^2+459*n+81)*x^3+(81*n^2+162*n)*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[(-(60*n - 150 + (135*n^2 - 261*n - 360)*x + (54*n^3 - 27*n^2 - 675*n - 162)*x^2 - (162*n^2 + 324*n)*x^3)*P[n - 1, x] + (25 + (75*n + 135)*x + (45*n^2 + 360*n + 207)*x^2 + (162*n^2 + 459*n + 81)*x^3 + (81*n^2 + 162*n)*x^4)*D[P[n - 1, x], x])/(30*n - 30 + (45*n^2 + 27*n - 72)*x + (27*n^3 + 27*n^2 - 54*n)*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] = (-(60*n - 150 + (135*n^2 - 261*n - 360)*x + (54*n^3 - 27*n^2 - 675*n - 162)*x^2 - (162*n^2 + 324*n)*x^3)*v[i - 1] + (25 + (75*n + 135)*x + (45*n^2 + 360*n + 207)*x^2 + (162*n^2 + 459*n + 81)*x^3 + (81*n^2 + 162*n)*x^4)*deriv(v[i - 1], x))/(30*n - 30 + (45*n^2 + 27*n - 72)*x + (27*n^3 + 27*n^2 - 54*n)*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-392f49b496f79fbf","steps":["basis A013610","\u03b8+1"],"edges":["e-74fcd2d5aef5a8cd","e-9e1b4f44bbd9ea65"],"text":"A026729 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A026729 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}