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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":"0ca90f4d7e1541e357fdad661700479a8205e6a4545b10d1d523ffe4121c4aaf","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-bf2f08bd0370af8d","role":"sequence","name":"basis_map_A013609 of Pascal-(1,4,1) array.","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:     2,       8;\n  2:     8,      64,      36;\n  3:    24,     288,     504,      128;\n  4:    80,    1280,    3600,     2560,      400;\n  5:   256,    5120,   20160,    25600,    10400,     1152;\n  6:   832,   19968,  100800,   184320,   134400,    36864,     3136;\n  7:  2688,   75264,  464256,  1116160,  1206400,   585216,   119168,    8192;\n  8:  8704,  278528, 2029824,  6053888,  8742400,  6340608,  2245376,  360448,   20736;\n  9: 28160, 1013760, 8524800, 30392320, 55180800, 54097920, 28600320, 7864320, 1036800, 51200;\n","recurrence_sentence":"The row polynomials satisfy (5+(30*n-5)*x+(55*n^2-5*n-7)*x^2+(30*n^3+25*n^2-17*n-7)*x^3+(5*n^4+10*n^3-5*n^2-4*n+2)*x^4-(2*n^3+2*n^2)*x^5)*P(n,x) = (20+(120*n+140)*x+(220*n^2+660*n+372)*x^2+(120*n^3+900*n^2+1332*n+468)*x^3+(20*n^4+360*n^3+1140*n^2+1128*n+280)*x^4+(40*n^4+232*n^3+408*n^2+248*n+48)*x^5-(16*n^3+64*n^2+80*n+32)*x^6)*P(n-2,x) + (10+(60*n+30)*x+(110*n^2+160*n+16)*x^2+(60*n^3+250*n^2+156*n-60)*x^3+(10*n^4+100*n^3+180*n^2+28*n-54)*x^4+(10*n^4+36*n^3+20*n^2-8*n-2)*x^5-(4*n^3+12*n^2+12*n+4)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((20 + (120*n + 140)*x + (220*n^2 + 660*n + 372)*x^2 + (120*n^3 + 900*n^2 + 1332*n + 468)*x^3 + (20*n^4 + 360*n^3 + 1140*n^2 + 1128*n + 280)*x^4 + (40*n^4 + 232*n^3 + 408*n^2 + 248*n + 48)*x^5 - (16*n^3 + 64*n^2 + 80*n + 32)*x^6)*P[n - 2, x] + (10 + (60*n + 30)*x + (110*n^2 + 160*n + 16)*x^2 + (60*n^3 + 250*n^2 + 156*n - 60)*x^3 + (10*n^4 + 100*n^3 + 180*n^2 + 28*n - 54)*x^4 + (10*n^4 + 36*n^3 + 20*n^2 - 8*n - 2)*x^5 - (4*n^3 + 12*n^2 + 12*n + 4)*x^6)*P[n - 1, x])/(5 + (30*n - 5)*x + (55*n^2 - 5*n - 7)*x^2 + (30*n^3 + 25*n^2 - 17*n - 7)*x^3 + (5*n^4 + 10*n^3 - 5*n^2 - 4*n + 2)*x^4 - (2*n^3 + 2*n^2)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((20 + (120*n + 140)*x + (220*n^2 + 660*n + 372)*x^2 + (120*n^3 + 900*n^2 + 1332*n + 468)*x^3 + (20*n^4 + 360*n^3 + 1140*n^2 + 1128*n + 280)*x^4 + (40*n^4 + 232*n^3 + 408*n^2 + 248*n + 48)*x^5 - (16*n^3 + 64*n^2 + 80*n + 32)*x^6)*v[i - 2] + (10 + (60*n + 30)*x + (110*n^2 + 160*n + 16)*x^2 + (60*n^3 + 250*n^2 + 156*n - 60)*x^3 + (10*n^4 + 100*n^3 + 180*n^2 + 28*n - 54)*x^4 + (10*n^4 + 36*n^3 + 20*n^2 - 8*n - 2)*x^5 - (4*n^3 + 12*n^2 + 12*n + 4)*x^6)*v[i - 1])/(5 + (30*n - 5)*x + (55*n^2 - 5*n - 7)*x^2 + (30*n^3 + 25*n^2 - 17*n - 7)*x^3 + (5*n^4 + 10*n^3 - 5*n^2 - 4*n + 2)*x^4 - (2*n^3 + 2*n^2)*x^5)); 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-ed9434bfd889c409","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-175e767cc5e2f6ad","e-e6c578432c7d29db"],"text":"A081579 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A081579 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}