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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":"c488076e6ca1ff15368837229e983447d721886819a98cc43ea0885687456b88","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-d7e00a1c4968fb6d","role":"sequence","name":"derivative of Triangular array of Motzkin polynomial coefficients.","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;\n  1:  0;\n  2:  1;\n  3:  3;\n  4:  6,   16;\n  5: 10,   80;\n  6: 15,  240,   135;\n  7: 21,  560,   945;\n  8: 28, 1120,  3780,  896;\n  9: 36, 2016, 11340, 8064;\n","recurrence_sentence":"The row polynomials satisfy (n^2-4+(2*n^4-10*n^3+2*n^2+40*n-40)*x+(n^6-6*n^5+9*n^4+12*n^3-48*n^2+48*n-16)*x^2)*P(n,x) = (n^2+2*n+(2*n^4-6*n^3+6*n^2-28*n)*x+(n^6-4*n^5+13*n^4-66*n^3+128*n^2-40*n)*x^2+(4*n^6-20*n^5+28*n^4-12*n^3)*x^3)*P(n-1,x) + (2*n*x+(4*n^3-12*n^2-4*n)*x^2+(2*n^5-4*n^4-14*n^3+48*n^2-16*n)*x^3-(8*n^5-16*n^4+8*n^3)*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n + (2*n^4 - 6*n^3 + 6*n^2 - 28*n)*x + (n^6 - 4*n^5 + 13*n^4 - 66*n^3 + 128*n^2 - 40*n)*x^2 + (4*n^6 - 20*n^5 + 28*n^4 - 12*n^3)*x^3)*P[n - 1, x] + (2*n*x + (4*n^3 - 12*n^2 - 4*n)*x^2 + (2*n^5 - 4*n^4 - 14*n^3 + 48*n^2 - 16*n)*x^3 - (8*n^5 - 16*n^4 + 8*n^3)*x^4)*D[P[n - 1, x], x])/(n^2 - 4 + (2*n^4 - 10*n^3 + 2*n^2 + 40*n - 40)*x + (n^6 - 6*n^5 + 9*n^4 + 12*n^3 - 48*n^2 + 48*n - 16)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((n^2 + 2*n + (2*n^4 - 6*n^3 + 6*n^2 - 28*n)*x + (n^6 - 4*n^5 + 13*n^4 - 66*n^3 + 128*n^2 - 40*n)*x^2 + (4*n^6 - 20*n^5 + 28*n^4 - 12*n^3)*x^3)*v[i - 1] + (2*n*x + (4*n^3 - 12*n^2 - 4*n)*x^2 + (2*n^5 - 4*n^4 - 14*n^3 + 48*n^2 - 16*n)*x^3 - (8*n^5 - 16*n^4 + 8*n^3)*x^4)*deriv(v[i - 1], x))/(n^2 - 4 + (2*n^4 - 10*n^3 + 2*n^2 + 40*n - 40)*x + (n^6 - 6*n^5 + 9*n^4 + 12*n^3 - 48*n^2 + 48*n - 16)*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-169bd483754ab390","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-07f016342004b27e","e-8501bcd693415561"],"text":"A055151 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A055151 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}