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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":"d856c41d73255580bc456b5fbdaef21de30e5d758f61be1ea15fc6f22bc580f1","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-a875e5d4789de94f","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","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:            256,             960,              675;\n  1:           4096,           23040,            32400,            13500;\n  2:          65536,          491520,          1036800,           864000,            253125;\n  3:        1048576,         9830400,         27648000,         34560000,          20250000,           4556250;\n  4:       16777216,       188743680,        663552000,       1105920000,         972000000,         437400000,          79734375;\n  5:      268435456,      3523215360,      14863564800,      30965760000,       36288000000,       24494400000,        8930250000,        1366875000;\n  6:     4294967296,     64424509440,     317089382400,     792723456000,     1161216000000,     1045094400000,      571536000000,      174960000000,       23066015625;\n  7:    68719476736,   1159641169920,    6522981580800,   19025362944000,    33443020800000,    37623398400000,    27433728000000,    12597120000000,     3321506250000,     384433593750;\n  8:  1099511627776,  20615843020800,  130459631616000,  434865438720000,   891813888000000,  1203948748800000,  1097349120000000,   671846400000000,   265720500000000,   61509375000000,    6343154296875;\n  9: 17592186044416, 362838837166080, 2551210573824000, 9567039651840000, 22422749184000000, 35315829964800000, 38626689024000000, 29561241600000000, 15588936000000000, 5412825000000000, 1116395156250000, 103797070312500;\n","recurrence_sentence":"The row polynomials satisfy (16+(15*n+30)*x)*P(n,x) = (256+(240*n+960)*x+(225*n+675)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 960*x + 675*x^2.","mathematica":"P[0, x_] := 256 + 960*x + 675*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((256 + (240*n + 960)*x + (225*n + 675)*x^2)*P[n - 1, x])/(16 + (15*n + 30)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 256 + 960*x + 675*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((256 + (240*n + 960)*x + (225*n + 675)*x^2)*v[i - 1])/(16 + (15*n + 30)*x)); 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-2bc1923905556bf6","steps":["basis A013610","\u03b8+1"],"edges":["e-fd6222a072627022","e-3f807a7aaf139f3f"],"text":"A038319 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038319 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}