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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":"390bacd2a0163ba6ea0b1792cd935f07061bf6e59cc3a273a94477a9d40466ef","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-e655e253465c7aff","role":"sequence","name":"veronese parent of Binomial coefficients C(n,k) with n-k odd, read by rows.","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: 0,  2,   3;\n  1: 1,  4,   3;\n  2: 0,  2,   9,  12,    5;\n  3: 1,  8,  18,  16,    5;\n  4: 0,  2,  15,  40,   50,   30,    7;\n  5: 1, 12,  45,  80,   75,   36,    7;\n  6: 0,  2,  21,  84,  175,  210,  147,  56,   9;\n  7: 1, 16,  84, 224,  350,  336,  196,  64,   9;\n  8: 0,  2,  27, 144,  420,  756,  882, 672, 324,  90, 11;\n  9: 1, 20, 135, 480, 1050, 1512, 1470, 960, 405, 100, 11;\n","recurrence_sentence":"The row polynomials satisfy (2+(2*n+4)*x+(n^2+3*n+2)*x^2)*P(n,x) = -(2*n+(3*n^2+5*n-4)*x+(n^3+3*n^2-8*n-14)*x^2-(6*n^2+22*n+16)*x^3-(n^3+6*n^2+11*n+6)*x^4)*P(n-1,x) + (1+(2*n+4)*x+(n^2+5*n+1)*x^2+(n^2-n-12)*x^3-(n^2+9*n+16)*x^4-(n^2+5*n+6)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2*x + 3*x^2.","mathematica":"P[0, x_] := 2*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + (3*n^2 + 5*n - 4)*x + (n^3 + 3*n^2 - 8*n - 14)*x^2 - (6*n^2 + 22*n + 16)*x^3 - (n^3 + 6*n^2 + 11*n + 6)*x^4)*P[n - 1, x] + (1 + (2*n + 4)*x + (n^2 + 5*n + 1)*x^2 + (n^2 - n - 12)*x^3 - (n^2 + 9*n + 16)*x^4 - (n^2 + 5*n + 6)*x^5)*D[P[n - 1, x], x])/(2 + (2*n + 4)*x + (n^2 + 3*n + 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n + (3*n^2 + 5*n - 4)*x + (n^3 + 3*n^2 - 8*n - 14)*x^2 - (6*n^2 + 22*n + 16)*x^3 - (n^3 + 6*n^2 + 11*n + 6)*x^4)*v[i - 1] + (1 + (2*n + 4)*x + (n^2 + 5*n + 1)*x^2 + (n^2 - n - 12)*x^3 - (n^2 + 9*n + 16)*x^4 - (n^2 + 5*n + 6)*x^5)*deriv(v[i - 1], x))/(2 + (2*n + 4)*x + (n^2 + 3*n + 2)*x^2)); 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-e655e253465c7aff","steps":["\u03b8+1"],"edges":["e-99be5c0e6cd30a3a"],"text":"Veronese parent of A109447 \u2192 \u03b8+1"},"headline":"Veronese parent of A109447 \u2192 \u03b8+1","chain":null}