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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":"a75b836489f332c0baa905ef4620bda34b1a68a5c965f925b2debe9d2c4bb43a","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-54fca3fdfcf03464","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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:             361,             532,              147;\n  1:            6859,           15162,             8379,             1372;\n  2:          130321,          384104,           318402,           104272,            12005;\n  3:         2476099,         9122470,         10082730,          4952920,          1140475,           100842;\n  4:        47045881,       207992316,        287357805,        188210960,         65007075,         11495988,          823543;\n  5:       893871739,      4610496338,       7643717613,       6258014420,       2881980325,        764483202,       109531219,         6588344;\n  6:     16983563041,    100113634768,     193640846196,     190243638368,     109515252350,      38733815568,      8324372644,      1001428288,       51883209;\n  7:    322687697779,   2139928943166,    4730369242788,    5421943693488,    3745421630370,    1655870615532,    474489240708,     85622118624,     8872028739,     403536070;\n  8:   6131066257801,  45176277689060,  112346269516215,  147167043108960,  118605018295050,   62923083390216,  22538238933630,   5422734179520,   842842730205,   76671853300,   3107227739;\n  9: 116490258898219, 944184203701354, 2608930036543215, 3844739001221580, 3541206974809350, 2191820738092524, 942098387425734, 283337860879920, 58718043537615, 8012208669850, 649410597451, 23727920916;\n","recurrence_sentence":"The row polynomials satisfy (19+(7*n+14)*x)*P(n,x) = (361+(133*n+532)*x+(49*n+147)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 532*x + 147*x^2.","mathematica":"P[0, x_] := 361 + 532*x + 147*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((361 + (133*n + 532)*x + (49*n + 147)*x^2)*P[n - 1, x])/(19 + (7*n + 14)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 361 + 532*x + 147*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((361 + (133*n + 532)*x + (49*n + 147)*x^2)*v[i - 1])/(19 + (7*n + 14)*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-5cf6acfdac607d6d","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-737551651b8dc504","e-353847383edc76b0"],"text":"A038333 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A038333 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}