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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":"203e2267da28e51238df11a7b5c77b6b7ee0b37a6c03115bcbda21f7b50a29dc","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-1b336857eed4ab2c","role":"sequence","name":"derivative of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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:   0;\n   2:   1;\n   3:   4,    4;\n   4:  10,   24,     9;\n   5:  20,   84,    72,    16;\n   6:  35,  224,   324,   160,    25;\n   7:  56,  504,  1080,   880,   300,    36;\n   8:  84, 1008,  2970,  3520,  1950,   504,   49;\n   9: 120, 1848,  7128, 11440,  9100,  3780,  784,   64;\n  10: 165, 3168, 15444, 32032, 34125, 20160, 6664, 1152, 81;\n","recurrence_sentence":"The row polynomials satisfy (4*n-8+(2*n^3-8*n^2+8*n)*x)*P(n,x) = (4*n+4+(2*n^3-2*n^2+4)*x+(n^3-n)*x^2)*P(n-1,x) + (8*x+(4*n^2-4*n+2)*x^2+(n^2-n)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n + 4 + (2*n^3 - 2*n^2 + 4)*x + (n^3 - n)*x^2)*P[n - 1, x] + (8*x + (4*n^2 - 4*n + 2)*x^2 + (n^2 - n)*x^3)*D[P[n - 1, x], x])/(4*n - 8 + (2*n^3 - 8*n^2 + 8*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((4*n + 4 + (2*n^3 - 2*n^2 + 4)*x + (n^3 - n)*x^2)*v[i - 1] + (8*x + (4*n^2 - 4*n + 2)*x^2 + (n^2 - n)*x^3)*deriv(v[i - 1], x))/(4*n - 8 + (2*n^3 - 8*n^2 + 8*n)*x)); 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-976cdb6e4a693ec8","steps":["D","\u03b8+1"],"edges":["e-1dbefba2139b076a","e-4452cad837de69a0"],"text":"A078812 \u2192 D \u2192 \u03b8+1"},"headline":"A078812 \u2192 D \u2192 \u03b8+1","chain":null}