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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":"dca463d17c7d974e95f11cd3ee988dcf17f764b08019f3b737016f9b5bf23adb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-4479b0484a11b442","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     2,      4;\n   3:     5,     24,       9;\n   4:    14,    112,     108,      16;\n   5:    42,    480,     810,     320,      25;\n   6:   132,   1980,    4950,    3520,     750,      36;\n   7:   429,   8008,   27027,   29120,   11375,    1512,      49;\n   8:  1430,  32032,  137592,  203840,  122500,   30240,    2744,     64;\n   9:  4862, 127296,  668304, 1279488, 1071000,  411264,   69972,   4608,   81;\n  10: 16796, 503880, 3139560, 7441920, 8139600, 4395384, 1173060, 145920, 7290, 100;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-6*n^3+2*n+(12*n^5-30*n^4+10*n^3+10*n^2-2*n)*x+(4*n^6-6*n^5+2*n^3)*x^2)*P(n,x) = (12*n^4-28*n^3+20*n^2-4*n+(43*n^5-108*n^4+23*n^3+110*n^2-84*n+16)*x+(15*n^6+19*n^5-153*n^4+131*n^3+62*n^2-102*n+28)*x^2+(12*n^6-8*n^5-24*n^4+24*n^3+6*n^2-14*n+4)*x^3)*P(n-1,x) + (n^3-n^2+(3*n^4-2*n^3-9*n^2+10*n-2)*x+(n^5+18*n^4-38*n^3+3*n^2+16*n-6)*x^2+(7*n^5-15*n^4+27*n^3+7*n^2-20*n+6)*x^3-(8*n^5+6*n^4-12*n^3+6*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n^4 - 28*n^3 + 20*n^2 - 4*n + (43*n^5 - 108*n^4 + 23*n^3 + 110*n^2 - 84*n + 16)*x + (15*n^6 + 19*n^5 - 153*n^4 + 131*n^3 + 62*n^2 - 102*n + 28)*x^2 + (12*n^6 - 8*n^5 - 24*n^4 + 24*n^3 + 6*n^2 - 14*n + 4)*x^3)*P[n - 1, x] + (n^3 - n^2 + (3*n^4 - 2*n^3 - 9*n^2 + 10*n - 2)*x + (n^5 + 18*n^4 - 38*n^3 + 3*n^2 + 16*n - 6)*x^2 + (7*n^5 - 15*n^4 + 27*n^3 + 7*n^2 - 20*n + 6)*x^3 - (8*n^5 + 6*n^4 - 12*n^3 + 6*n - 2)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 6*n^3 + 2*n + (12*n^5 - 30*n^4 + 10*n^3 + 10*n^2 - 2*n)*x + (4*n^6 - 6*n^5 + 2*n^3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((12*n^4 - 28*n^3 + 20*n^2 - 4*n + (43*n^5 - 108*n^4 + 23*n^3 + 110*n^2 - 84*n + 16)*x + (15*n^6 + 19*n^5 - 153*n^4 + 131*n^3 + 62*n^2 - 102*n + 28)*x^2 + (12*n^6 - 8*n^5 - 24*n^4 + 24*n^3 + 6*n^2 - 14*n + 4)*x^3)*v[i - 1] + (n^3 - n^2 + (3*n^4 - 2*n^3 - 9*n^2 + 10*n - 2)*x + (n^5 + 18*n^4 - 38*n^3 + 3*n^2 + 16*n - 6)*x^2 + (7*n^5 - 15*n^4 + 27*n^3 + 7*n^2 - 20*n + 6)*x^3 - (8*n^5 + 6*n^4 - 12*n^3 + 6*n - 2)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 6*n^3 + 2*n + (12*n^5 - 30*n^4 + 10*n^3 + 10*n^2 - 2*n)*x + (4*n^6 - 6*n^5 + 2*n^3)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4479b0484a11b442","steps":["(\u03b8+1)\u00b2"],"edges":["e-f31b0ac2185e97c9"],"text":"A120986 \u2192 (\u03b8+1)\u00b2"},"headline":"A120986 \u2192 (\u03b8+1)\u00b2","chain":null}