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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":"c555a34d000bddc9bbb8eb4af8d419b16ab9db4f9965e769597dd4784d41756f","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-7d2bbb701a5b14c4","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^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:             400,             3520,               4356;\n  1:            8000,           105600,             261360,             170368;\n  2:          160000,          2816000,           10454400,           13629440,             5856400;\n  3:         3200000,         70400000,          348480000,          681472000,           585640000,           185530752;\n  4:        64000000,       1689600000,        10454400000,        27258880000,         35138400000,         22263690240,          5555615296;\n  5:      1280000000,      39424000000,       292723200000,       954060800000,       1639792000000,       1558458316800,        777786141440,        159638904832;\n  6:     25600000000,     901120000000,      7805952000000,     30529945600000,      65591680000000,      83117776896000,      62222891315200,      25542224773120,       4444945756416;\n  7:    512000000000,   20275200000000,    200724480000000,    915898368000000,    2361300480000000,    3740299960320000,    3733373478912000,    2298800229580800,     800090236154880,     120726921779200;\n  8:  10240000000000,  450560000000000,   5018112000000000,  26168524800000000,   78710016000000000,  149611998412800000,  186668673945600000,  153253348638720000,   80009023615488000,   24145384355840000,   3213750657762304;\n  9: 204800000000000, 9912320000000000, 122664960000000000, 719634432000000000, 2473743360000000000, 5485773275136000000, 8213421653606400000, 8428934175129600000, 5867328398469120000, 2655992279142400000, 707025144707706880, 84141835403231232;\n","recurrence_sentence":"The row polynomials satisfy (100+(330*n+550)*x+(121*n^2+484*n+484)*x^2)*P(n,x) = (2000+(6600*n+19800)*x+(2420*n^2+21780*n+41140)*x^2+(2662*n^2+15972*n+23958)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 3520*x + 4356*x^2.","mathematica":"P[0, x_] := 400 + 3520*x + 4356*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2000 + (6600*n + 19800)*x + (2420*n^2 + 21780*n + 41140)*x^2 + (2662*n^2 + 15972*n + 23958)*x^3)*P[n - 1, x])/(100 + (330*n + 550)*x + (121*n^2 + 484*n + 484)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 400 + 3520*x + 4356*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2000 + (6600*n + 19800)*x + (2420*n^2 + 21780*n + 41140)*x^2 + (2662*n^2 + 15972*n + 23958)*x^3)*v[i - 1])/(100 + (330*n + 550)*x + (121*n^2 + 484*n + 484)*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-cfc5b48d32898a3e","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-21c2803b3d0372c9","e-ad962e8ac3cd2efe"],"text":"A038301 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038301 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}