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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":"31225943739e73d5ede50d0a4383135c34db2b0a9c0e8a8509b92479b6ce68f8","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-32899adace0e98a1","role":"sequence","name":"basis_map_A013611 of A Jacobsthal triangle.","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:   3,    32,     48;\n  3:   5,    88,    336,     256;\n  4:  11,   240,   1344,    2560,    1280;\n  5:  21,   584,   4512,   13824,   16640,     6144;\n  6:  43,  1408,  13584,   57344,  113920,    98304,    28672;\n  7:  85,  3256,  38064,  205568,  572160,   817152,   544768,   131072;\n  8: 171,  7440, 101664,  668160, 2401280,  4853760,  5332992,  2883584,   589824;\n  9: 341, 16680, 261504, 2027520, 8942080, 23506944, 36700160, 32505856, 14745600, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (72*n+(144*n^2+368*n)*x+(128*n^3+448*n^2+320*n)*x^2)*P(n,x) = -(288*n-360+(864*n^2+320*n-2272)*x+(512*n^3+1152*n^2-2688*n-3968)*x^2-(1024*n^2+3584*n+2560)*x^3)*P(n-1,x) + (81+(324*n+864)*x+(288*n^2+2448*n+3104)*x^2+(1280*n^2+5056*n+4096)*x^3+(512*n^2+1792*n+1280)*x^4)*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[(-(288*n - 360 + (864*n^2 + 320*n - 2272)*x + (512*n^3 + 1152*n^2 - 2688*n - 3968)*x^2 - (1024*n^2 + 3584*n + 2560)*x^3)*P[n - 1, x] + (81 + (324*n + 864)*x + (288*n^2 + 2448*n + 3104)*x^2 + (1280*n^2 + 5056*n + 4096)*x^3 + (512*n^2 + 1792*n + 1280)*x^4)*D[P[n - 1, x], x])/(72*n + (144*n^2 + 368*n)*x + (128*n^3 + 448*n^2 + 320*n)*x^2)]];\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] = (-(288*n - 360 + (864*n^2 + 320*n - 2272)*x + (512*n^3 + 1152*n^2 - 2688*n - 3968)*x^2 - (1024*n^2 + 3584*n + 2560)*x^3)*v[i - 1] + (81 + (324*n + 864)*x + (288*n^2 + 2448*n + 3104)*x^2 + (1280*n^2 + 5056*n + 4096)*x^3 + (512*n^2 + 1792*n + 1280)*x^4)*deriv(v[i - 1], x))/(72*n + (144*n^2 + 368*n)*x + (128*n^3 + 448*n^2 + 320*n)*x^2)); 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-ba196dcd01800741","steps":["basis A013611","\u03b8+1"],"edges":["e-4cbbf4b06abf62f6","e-b6e7fe7aac5a9a7a"],"text":"A113953 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A113953 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}