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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":"557be528fb8ac819a2c60571b7aea89400170fd6efcd23eb7512177077bf31b1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1852b727a5d6b55d","role":"sequence","name":"basis_map_A013609 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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;\n  2:    2,     4;\n  3:    5,    20,     12;\n  4:   13,    76,     96,     32;\n  5:   34,   260,    504,    352,     80;\n  6:   89,   840,   2196,   2368,   1120,    192;\n  7:  233,  2616,   8604,  12608,   9200,   3264,    448;\n  8:  610,  7940,  31464,  58400,  58000,  31680,   8960,  1024;\n  9: 1597, 23644, 109608, 246496, 310800, 230976, 100352, 23552, 2304;\n","recurrence_sentence":"The row polynomials satisfy -(40*n-60+(64*n-96)*x-(16*n^3-72*n^2+104*n-48)*x^2)*P(n,x) = -(60*n-80+(20*n^2+116*n-168)*x-(24*n^3-104*n^2+16*n+64)*x^2-(16*n^3-48*n^2+32*n)*x^3)*P(n-1,x) - (25+140*x-(20*n^2-40*n-212)*x^2-(48*n^2-96*n-64)*x^3-(16*n^2-32*n)*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[(-(60*n - 80 + (20*n^2 + 116*n - 168)*x - (24*n^3 - 104*n^2 + 16*n + 64)*x^2 - (16*n^3 - 48*n^2 + 32*n)*x^3)*P[n - 1, x] - (25 + 140*x - (20*n^2 - 40*n - 212)*x^2 - (48*n^2 - 96*n - 64)*x^3 - (16*n^2 - 32*n)*x^4)*D[P[n - 1, x], x])/(-40*n + 60 - (64*n - 96)*x + (16*n^3 - 72*n^2 + 104*n - 48)*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] = (-(60*n - 80 + (20*n^2 + 116*n - 168)*x - (24*n^3 - 104*n^2 + 16*n + 64)*x^2 - (16*n^3 - 48*n^2 + 32*n)*x^3)*v[i - 1] - (25 + 140*x - (20*n^2 - 40*n - 212)*x^2 - (48*n^2 - 96*n - 64)*x^3 - (16*n^2 - 32*n)*x^4)*deriv(v[i - 1], x))/(-40*n + 60 - (64*n - 96)*x + (16*n^3 - 72*n^2 + 104*n - 48)*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-0eba401b117f16a9","steps":["basis A013609","\u03b8+1"],"edges":["e-354781356c20418d","e-7e6beaa89e0e6fea"],"text":"A085478 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A085478 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}