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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":"ab0cb88884de8a76e6c5f8ee0f5688d49238de583be2f2d1eaad816f6f2dcf0c","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-344e4bfa9933088a","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(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:          144,           1056,            1089;\n  1:         1728,          19008,           39204,            21296;\n  2:        20736,         304128,          940896,          1022208,           366025;\n  3:       248832,        4561920,        18817920,         30666240,         21961500,          5797836;\n  4:      2985984,       65691648,       338722560,        735989760,        790614000,        417444192,         86806489;\n  5:     35831808,      919683072,      5690539008,      15455784960,      22137192000,      17532656064,       7291745076,       1247178944;\n  6:    429981696,    12612796416,     91048624128,     296751071232,     531292608000,     561044994048,     350003763648,     119729178624,      17363069361;\n  7:   5159780352,   170272751616,   1404750200832,    5341519282176,   11475920332800,   15148214839296,   12600135491328,    6465375645696,    1875211490988,     235794769100;\n  8:  61917364224,  2270303354880,  21071253012480,   91568901980160,  229518406656000,  363557156143104,  378004064739840,  258615025827840,  112512689459280,   28295372292000,   3138428376721;\n  9: 743008370688, 29968004284416, 309045044183040, 1510886882672640, 4328061382656000, 7998257435148288, 9979307309131776, 8534295852318720, 4950558336208320, 1867494571272000, 414272545727172, 41084880567984;\n","recurrence_sentence":"The row polynomials satisfy (144+(396*n+660)*x+(121*n^2+484*n+484)*x^2)*P(n,x) = (1728+(4752*n+14256)*x+(1452*n^2+13068*n+24684)*x^2+(1331*n^2+7986*n+11979)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 1056*x + 1089*x^2.","mathematica":"P[0, x_] := 144 + 1056*x + 1089*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((1728 + (4752*n + 14256)*x + (1452*n^2 + 13068*n + 24684)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*P[n - 1, x])/(144 + (396*n + 660)*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] = 144 + 1056*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1728 + (4752*n + 14256)*x + (1452*n^2 + 13068*n + 24684)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*v[i - 1])/(144 + (396*n + 660)*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-344e4bfa9933088a","steps":["(\u03b8+1)\u00b2"],"edges":["e-c64fa5f58bac9310"],"text":"A038337 \u2192 (\u03b8+1)\u00b2"},"headline":"A038337 \u2192 (\u03b8+1)\u00b2","chain":null}