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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":"3cf1e1db9782bdb124e9a64da269c4adcbaad52516e86d5b832b6c289450f918","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-353fb5a029e4e669","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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: 1,   8;\n  2: 1,  20,   18;\n  3: 1,  36,   81,    32;\n  4: 1,  56,  225,   208,    50;\n  5: 1,  80,  495,   784,   425,    72;\n  6: 1, 108,  945,  2240,  2025,   756,    98;\n  7: 1, 140, 1638,  5376,  7125,  4356,  1225,   128;\n  8: 1, 176, 2646, 11424, 20625, 18216,  8281,  1856,  162;\n  9: 1, 216, 4050, 22176, 51975, 61776, 40131, 14400, 2673, 200;\n","recurrence_sentence":"The row polynomials satisfy -(72*n+36-(44*n^3+66*n^2+346*n-438)*x+(2*n^5+5*n^4+116*n^3-395*n^2+356*n-84)*x^2-(4*n^5-20*n^4+32*n^3-16*n^2)*x^3)*P(n,x) = -(72*n+36-(44*n^3+66*n^2+202*n-1158)*x+(2*n^5+5*n^4+78*n^3-661*n^2-164*n+596)*x^2-(3*n^5-29*n^4-64*n^3+204*n^2-2*n-112)*x^3-(2*n^5-6*n^4-2*n^3+14*n^2-8)*x^4)*P(n-1,x) - (144*x-(88*n^2+176*n+988)*x^2+(4*n^4+16*n^3+250*n^2-348*n-568)*x^3-(7*n^4-20*n^3-92*n^2+108*n+110)*x^4-(2*n^4-4*n^3-6*n^2+8*n+8)*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[(-(72*n + 36 - (44*n^3 + 66*n^2 + 202*n - 1158)*x + (2*n^5 + 5*n^4 + 78*n^3 - 661*n^2 - 164*n + 596)*x^2 - (3*n^5 - 29*n^4 - 64*n^3 + 204*n^2 - 2*n - 112)*x^3 - (2*n^5 - 6*n^4 - 2*n^3 + 14*n^2 - 8)*x^4)*P[n - 1, x] - (144*x - (88*n^2 + 176*n + 988)*x^2 + (4*n^4 + 16*n^3 + 250*n^2 - 348*n - 568)*x^3 - (7*n^4 - 20*n^3 - 92*n^2 + 108*n + 110)*x^4 - (2*n^4 - 4*n^3 - 6*n^2 + 8*n + 8)*x^5)*D[P[n - 1, x], x])/(-72*n - 36 + (44*n^3 + 66*n^2 + 346*n - 438)*x - (2*n^5 + 5*n^4 + 116*n^3 - 395*n^2 + 356*n - 84)*x^2 + (4*n^5 - 20*n^4 + 32*n^3 - 16*n^2)*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] = (-(72*n + 36 - (44*n^3 + 66*n^2 + 202*n - 1158)*x + (2*n^5 + 5*n^4 + 78*n^3 - 661*n^2 - 164*n + 596)*x^2 - (3*n^5 - 29*n^4 - 64*n^3 + 204*n^2 - 2*n - 112)*x^3 - (2*n^5 - 6*n^4 - 2*n^3 + 14*n^2 - 8)*x^4)*v[i - 1] - (144*x - (88*n^2 + 176*n + 988)*x^2 + (4*n^4 + 16*n^3 + 250*n^2 - 348*n - 568)*x^3 - (7*n^4 - 20*n^3 - 92*n^2 + 108*n + 110)*x^4 - (2*n^4 - 4*n^3 - 6*n^2 + 8*n + 8)*x^5)*deriv(v[i - 1], x))/(-72*n - 36 + (44*n^3 + 66*n^2 + 346*n - 438)*x - (2*n^5 + 5*n^4 + 116*n^3 - 395*n^2 + 356*n - 84)*x^2 + (4*n^5 - 20*n^4 + 32*n^3 - 16*n^2)*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-353fb5a029e4e669","steps":["(\u03b8+1)\u00b2"],"edges":["e-6f5daa3d60a537b3"],"text":"A207607 \u2192 (\u03b8+1)\u00b2"},"headline":"A207607 \u2192 (\u03b8+1)\u00b2","chain":null}