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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":"a578a19d0569d4e3e8bfde39361a5c924f0f5c9d13ce3ac1039d9898adde0a26","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-32cf9641286d8e9f","role":"sequence","name":"basis_map_A007318 of 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:            324,             504,              147;\n  1:           5832,           13608,             7938,             1372;\n  2:         104976,          326592,           285768,            98784,            12005;\n  3:        1889568,         7348320,          8573040,          4445280,          1080450,           100842;\n  4:       34012224,       158723712,        231472080,        160030080,         58344300,         10890936,          823543;\n  5:      612220032,      3333197952,       5833096416,       5040947520,       2450460600,        686128968,       103766418,         6588344;\n  6:    11019960576,     68568643584,     139994313984,     145179288576,      88216581600,      32934190464,      7471182096,       948721536,       51883209;\n  7:   198359290368,   1388515032576,    3239868409344,    3919840791552,    2858217243840,    1333834713792,    403443833184,     76846444416,     8405079858,     403536070;\n  8:  3570467226624,  27770300651520,   72897039210240,  100795906068480,   85746517315200,   48018049696512,  18154972493280,   4610786664960,   756457187220,   72636492600,   3107227739;\n  9: 64268410079232, 549851952900096, 1603734862625280, 2494698675194880, 2425401489772800, 1584595639984896, 718936910733888, 228233939915520, 49926174356520, 7191012767400, 615231092322, 23727920916;\n","recurrence_sentence":"The row polynomials satisfy (18+(7*n+14)*x)*P(n,x) = (324+(126*n+504)*x+(49*n+147)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 504*x + 147*x^2.","mathematica":"P[0, x_] := 324 + 504*x + 147*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((324 + (126*n + 504)*x + (49*n + 147)*x^2)*P[n - 1, x])/(18 + (7*n + 14)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 324 + 504*x + 147*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((324 + (126*n + 504)*x + (49*n + 147)*x^2)*v[i - 1])/(18 + (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-7dcf11eddf2b9a8a","steps":["basis A007318","\u03b8+1"],"edges":["e-5a47242ff4b92549","e-ad3081c48ad57b30"],"text":"A038321 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A038321 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}