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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":"e09c58fcef7db8eda5184968a01de79b4134ca32c49dd457d4272255606da437","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-6b8ac0ed9d98a614","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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,            4;\n  2:           8,           40,            27;\n  3:          56,          472,           729,           288;\n  4:         616,         7432,         18639,         16128,           4500;\n  5:        9856,       155872,        549054,        755088,         450000,          97200;\n  6:      216832,      4257088,      19444140,      37109952,       34676325,       15746400,        2778300;\n  7:     6288128,    147740736,     832076604,    2044074688,     2629173825,     1854795024,      680683500,     101606400;\n  8:   232660736,   6371897664,   42753833964,  128883666112,   212258632725,   204923786976,   116070245676,   35765452800,    4629441600;\n  9: 10702393856, 334986225024, 2611831000824, 9348955358272, 18826029878850, 23180853601476, 17890813253376, 8467298737920, 2249908617600, 257191200000;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(4*n^5-4*n^4+4*n^3+4*n^2-8*n)*x-(9*n^5-9*n^4-9*n^3+9*n^2)*x^2)*P(n-2,x) + ((4*n^5-4*n^4+4*n^3+4*n^2-8*n)*x^2+(4*n^5-4*n^4-4*n^3+4*n^2)*x^3)*d/dx P(n-2,x) + (4*n^3+4*n+(8*n^3-8*n)*x)*P(n-1,x) - (8*n^3-8*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x - (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + ((4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x^2 + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^3)*D[P[n - 2, x], x] + (4*n^3 + 4*n + (8*n^3 - 8*n)*x)*P[n - 1, x] - (8*n^3 - 8*n)*x^2*D[P[n - 1, x], x])/(4*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x - (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*v[i - 2] + ((4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x^2 + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^3)*deriv(v[i - 2], x) + (4*n^3 + 4*n + (8*n^3 - 8*n)*x)*v[i - 1] - (8*n^3 - 8*n)*x^2*deriv(v[i - 1], x))/(4*n - 4)); 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-ecd77b030b8fab94","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-8bd02eafb0f0a53d","e-f669a93fd3b4e549"],"text":"A128813 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A128813 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}