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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":"08a9037a8f8f5294b8e1619bbf128cea64bc2cd9ff372df97524d3c11746d3f5","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-7dcf11eddf2b9a8a","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(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:          121,           616,            441;\n  1:         1331,         10164,          14553,            5488;\n  2:        14641,        149072,         320166,          241472,           60025;\n  3:       161051,       2049740,        5869710,         6640480,         3301375,          605052;\n  4:      1771561,      27056568,       96850215,       146090560,       108945375,        39933432,         5764801;\n  5:     19487171,     347225956,     1491493311,      2812243280,      2796264625,      1537437132,       443889677,        52706752;\n  6:    214358881,    4365126304,    21875235228,     49495481728,     61517821750,     45098155872,     19531145788,      4638194176,       466948881;\n  7:   2357947691,   54018438012,   309378326796,    816675448512,   1218052870650,   1116179357832,    644527811004,    229590611712,     46227939219,     4035360700;\n  8:  25937424601,  660225353480,  4253951993445,  12833471333760,  22330969295250,  24555945872304,  17724514802610,   8418322429440,   2542536657045,   443889677000,   34179505129;\n  9: 285311670611, 7988726777108, 57192021245205, 194106253923120, 386006754960750, 495211575091464, 428933258223162, 254654253490560, 102548978500815, 26855325458500, 4135720120609, 284735050992;\n","recurrence_sentence":"The row polynomials satisfy (121+(231*n+385)*x+(49*n^2+196*n+196)*x^2)*P(n,x) = (1331+(2541*n+7623)*x+(539*n^2+4851*n+9163)*x^2+(343*n^2+2058*n+3087)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 616*x + 441*x^2.","mathematica":"P[0, x_] := 121 + 616*x + 441*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((1331 + (2541*n + 7623)*x + (539*n^2 + 4851*n + 9163)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*P[n - 1, x])/(121 + (231*n + 385)*x + (49*n^2 + 196*n + 196)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 121 + 616*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1331 + (2541*n + 7623)*x + (539*n^2 + 4851*n + 9163)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*v[i - 1])/(121 + (231*n + 385)*x + (49*n^2 + 196*n + 196)*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-7dcf11eddf2b9a8a","steps":["(\u03b8+1)\u00b2"],"edges":["e-0b1753c1c25fcbb3"],"text":"A038321 \u2192 (\u03b8+1)\u00b2"},"headline":"A038321 \u2192 (\u03b8+1)\u00b2","chain":null}