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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":"ddf1e02e92863ea7e194a5303e193ff931d95729641ea81e62cd0d8b03db13a6","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-f07088d208415183","role":"sequence","name":"basis_map_A013609 of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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:      4,       16;\n  2:     18,      144,       144;\n  3:     80,      960,      2016,      1024;\n  4:    356,     5696,     18288,     19456,       6400;\n  5:   1584,    31680,    137088,    224256,     153600,      36864;\n  6:   7048,   169152,    921312,   2040832,    2150400,    1069056,     200704;\n  7:  31360,   878080,   5766912,  16138240,   23065600,   17620992,    6823936,   1048576;\n  8: 139536,  4465152,  34335360, 116195328,  210182400,  217866240,  129454080,  40894464,   5308416;\n  9: 620864, 22351104, 196964352, 782229504, 1713945600, 2250215424, 1813561344, 877658112, 233570304, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (3+(18*n-3)*x+(33*n^2-3*n-7)*x^2+(18*n^3+15*n^2-27*n)*x^3+(3*n^4+6*n^3-17*n^2-8*n+4)*x^4-(4*n^3+4*n^2)*x^5)*P(n,x) = (6+(36*n+42)*x+(66*n^2+198*n+106)*x^2+(36*n^3+270*n^2+366*n+104)*x^3+(6*n^4+108*n^3+314*n^2+204*n)*x^4+(12*n^4+64*n^3+44*n^2-88*n-64)*x^5-(16*n^3+64*n^2+80*n+32)*x^6)*P(n-2,x) + (12+(72*n+36)*x+(132*n^2+192*n+8)*x^2+(72*n^3+300*n^2+120*n-100)*x^3+(12*n^4+120*n^3+160*n^2-112*n-132)*x^4+(12*n^4+32*n^3-60*n^2-144*n-64)*x^5-(16*n^3+48*n^2+48*n+16)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 16*x;\nP[n_, x_] := P[n, x] = Expand[Together[((6 + (36*n + 42)*x + (66*n^2 + 198*n + 106)*x^2 + (36*n^3 + 270*n^2 + 366*n + 104)*x^3 + (6*n^4 + 108*n^3 + 314*n^2 + 204*n)*x^4 + (12*n^4 + 64*n^3 + 44*n^2 - 88*n - 64)*x^5 - (16*n^3 + 64*n^2 + 80*n + 32)*x^6)*P[n - 2, x] + (12 + (72*n + 36)*x + (132*n^2 + 192*n + 8)*x^2 + (72*n^3 + 300*n^2 + 120*n - 100)*x^3 + (12*n^4 + 120*n^3 + 160*n^2 - 112*n - 132)*x^4 + (12*n^4 + 32*n^3 - 60*n^2 - 144*n - 64)*x^5 - (16*n^3 + 48*n^2 + 48*n + 16)*x^6)*P[n - 1, x])/(3 + (18*n - 3)*x + (33*n^2 - 3*n - 7)*x^2 + (18*n^3 + 15*n^2 - 27*n)*x^3 + (3*n^4 + 6*n^3 - 17*n^2 - 8*n + 4)*x^4 - (4*n^3 + 4*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] = 4 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = ((6 + (36*n + 42)*x + (66*n^2 + 198*n + 106)*x^2 + (36*n^3 + 270*n^2 + 366*n + 104)*x^3 + (6*n^4 + 108*n^3 + 314*n^2 + 204*n)*x^4 + (12*n^4 + 64*n^3 + 44*n^2 - 88*n - 64)*x^5 - (16*n^3 + 64*n^2 + 80*n + 32)*x^6)*v[i - 2] + (12 + (72*n + 36)*x + (132*n^2 + 192*n + 8)*x^2 + (72*n^3 + 300*n^2 + 120*n - 100)*x^3 + (12*n^4 + 120*n^3 + 160*n^2 - 112*n - 132)*x^4 + (12*n^4 + 32*n^3 - 60*n^2 - 144*n - 64)*x^5 - (16*n^3 + 48*n^2 + 48*n + 16)*x^6)*v[i - 1])/(3 + (18*n - 3)*x + (33*n^2 - 3*n - 7)*x^2 + (18*n^3 + 15*n^2 - 27*n)*x^3 + (3*n^4 + 6*n^3 - 17*n^2 - 8*n + 4)*x^4 - (4*n^3 + 4*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-1bffaa6198344f93","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-c90acfb06fc8929f","e-5c6c45b8bc6280a1"],"text":"A059473 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A059473 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}