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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":"7025a95682c1eb442d442c1cda1c1bc8eafeae00470b7feda41324e3aa5c581c","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-0892e4c474e4320d","role":"sequence","name":"basis_map_A013610 of Riordan array (1, 2+2x).","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,     24;\n  2:    6,    120,      324;\n  3:   16,    480,     2592,      3456;\n  4:   44,   1728,    13932,     38016,     32400;\n  5:  120,   5856,    62856,    259200,    453600,     279936;\n  6:  328,  19104,   255960,   1413504,   3758400,    4758912,    2286144;\n  7:  896,  60672,   974592,   6746112,  24105600,   46469376,   45722880,   17915904;\n  8: 2448, 188928,  3538080,  29445120, 132224400,  343201536,  514382400,  412065792,  136048896;\n  9: 6688, 579456, 12394944, 120517632, 651952800, 2130033024, 4302523008, 5249359872, 3537271296, 1007769600;\n","recurrence_sentence":"The row polynomials satisfy (4+(36*n-4)*x+(78*n^2+30*n-32)*x^2+(54*n^3+96*n^2-150*n)*x^3+(9*n^4+54*n^3-153*n^2-198*n+72)*x^4-(54*n^3+162*n^2)*x^5)*P(n,x) = (8+(72*n+88)*x+(156*n^2+588*n+344)*x^2+(108*n^3+984*n^2+1524*n+456)*x^3+(18*n^4+504*n^3+1674*n^2+612*n-432)*x^4+(54*n^4+432*n^3-270*n^2-2808*n-1728)*x^5-(324*n^3+1944*n^2+2916*n+1296)*x^6)*P(n-2,x) + (8+(72*n+88)*x+(156*n^2+504*n+260)*x^2+(108*n^3+876*n^2+900*n-60)*x^3+(18*n^4+468*n^3+1134*n^2-756*n-1296)*x^4+(54*n^4+324*n^3-810*n^2-3348*n-1836)*x^5-(324*n^3+1620*n^2+2268*n+972)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 24*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 24*x;\nP[n_, x_] := P[n, x] = Expand[Together[((8 + (72*n + 88)*x + (156*n^2 + 588*n + 344)*x^2 + (108*n^3 + 984*n^2 + 1524*n + 456)*x^3 + (18*n^4 + 504*n^3 + 1674*n^2 + 612*n - 432)*x^4 + (54*n^4 + 432*n^3 - 270*n^2 - 2808*n - 1728)*x^5 - (324*n^3 + 1944*n^2 + 2916*n + 1296)*x^6)*P[n - 2, x] + (8 + (72*n + 88)*x + (156*n^2 + 504*n + 260)*x^2 + (108*n^3 + 876*n^2 + 900*n - 60)*x^3 + (18*n^4 + 468*n^3 + 1134*n^2 - 756*n - 1296)*x^4 + (54*n^4 + 324*n^3 - 810*n^2 - 3348*n - 1836)*x^5 - (324*n^3 + 1620*n^2 + 2268*n + 972)*x^6)*P[n - 1, x])/(4 + (36*n - 4)*x + (78*n^2 + 30*n - 32)*x^2 + (54*n^3 + 96*n^2 - 150*n)*x^3 + (9*n^4 + 54*n^3 - 153*n^2 - 198*n + 72)*x^4 - (54*n^3 + 162*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 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = ((8 + (72*n + 88)*x + (156*n^2 + 588*n + 344)*x^2 + (108*n^3 + 984*n^2 + 1524*n + 456)*x^3 + (18*n^4 + 504*n^3 + 1674*n^2 + 612*n - 432)*x^4 + (54*n^4 + 432*n^3 - 270*n^2 - 2808*n - 1728)*x^5 - (324*n^3 + 1944*n^2 + 2916*n + 1296)*x^6)*v[i - 2] + (8 + (72*n + 88)*x + (156*n^2 + 504*n + 260)*x^2 + (108*n^3 + 876*n^2 + 900*n - 60)*x^3 + (18*n^4 + 468*n^3 + 1134*n^2 - 756*n - 1296)*x^4 + (54*n^4 + 324*n^3 - 810*n^2 - 3348*n - 1836)*x^5 - (324*n^3 + 1620*n^2 + 2268*n + 972)*x^6)*v[i - 1])/(4 + (36*n - 4)*x + (78*n^2 + 30*n - 32)*x^2 + (54*n^3 + 96*n^2 - 150*n)*x^3 + (9*n^4 + 54*n^3 - 153*n^2 - 198*n + 72)*x^4 - (54*n^3 + 162*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-f5a001953ee28e8f","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-937d6a83cea8d959","e-5d06a39769958677"],"text":"A099040 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A099040 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}