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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":"65637ccec8337b4dc1e43dd94be0d9e3b1faac1bedb9778ff15d54af5e3469a7","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-59c36108e7f99c2f","role":"sequence","name":"basis_map_A013610 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:     1;\n  1:     3,      6;\n  2:    10,     42,      27;\n  3:    33,    210,     297,      108;\n  4:   109,    930,    2052,     1620,      405;\n  5:   360,   3852,   11583,    14364,     7695,     1458;\n  6:  1189,  15300,   58320,    99252,    83430,    33534,     5103;\n  7:  3927,  59046,  272727,   591732,   684045,   430110,   137781,   17496;\n  8: 12970, 223134, 1211058,  3198636,  4726755,  4086774,  2041200,  542376,   59049;\n  9: 42837, 829830, 5175711, 16122780, 29078190, 32169312, 22070475, 9115416, 2066715, 196830;\n","recurrence_sentence":"The row polynomials satisfy -(78*n+(117*n^2-189*n)*x+(81*n^3-459*n^2-972*n)*x^2-(243*n^3+648*n^2+405*n)*x^3)*P(n,x) = -(624*n-390+(1287*n^2+1881*n-2916)*x+(648*n^3+3294*n^2-1188*n-7236)*x^2+(972*n^3+405*n^2-6561*n-6804)*x^3-(1458*n^2+3888*n+2430)*x^4)*P(n-1,x) + (169+(507*n+1599)*x+(351*n^2+3627*n+5598)*x^2+(1863*n^2+8829*n+8802)*x^3+(2673*n^2+8181*n+5913)*x^4+(729*n^2+1944*n+1215)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(624*n - 390 + (1287*n^2 + 1881*n - 2916)*x + (648*n^3 + 3294*n^2 - 1188*n - 7236)*x^2 + (972*n^3 + 405*n^2 - 6561*n - 6804)*x^3 - (1458*n^2 + 3888*n + 2430)*x^4)*P[n - 1, x] + (169 + (507*n + 1599)*x + (351*n^2 + 3627*n + 5598)*x^2 + (1863*n^2 + 8829*n + 8802)*x^3 + (2673*n^2 + 8181*n + 5913)*x^4 + (729*n^2 + 1944*n + 1215)*x^5)*D[P[n - 1, x], x])/(-78*n - (117*n^2 - 189*n)*x - (81*n^3 - 459*n^2 - 972*n)*x^2 + (243*n^3 + 648*n^2 + 405*n)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(624*n - 390 + (1287*n^2 + 1881*n - 2916)*x + (648*n^3 + 3294*n^2 - 1188*n - 7236)*x^2 + (972*n^3 + 405*n^2 - 6561*n - 6804)*x^3 - (1458*n^2 + 3888*n + 2430)*x^4)*v[i - 1] + (169 + (507*n + 1599)*x + (351*n^2 + 3627*n + 5598)*x^2 + (1863*n^2 + 8829*n + 8802)*x^3 + (2673*n^2 + 8181*n + 5913)*x^4 + (729*n^2 + 1944*n + 1215)*x^5)*deriv(v[i - 1], x))/(-78*n - (117*n^2 - 189*n)*x - (81*n^3 - 459*n^2 - 972*n)*x^2 + (243*n^3 + 648*n^2 + 405*n)*x^3)); 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-d5c14b9fec875ecc","steps":["basis A013610","\u03b8+1"],"edges":["e-16250729b41a60db","e-511c987279233666"],"text":"A121574 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A121574 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}