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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":"61f47c108cc4afc4a52a52df95fc5ae20e81538541ce4a4fd1e1ad01fa4f0806","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6d8c9ba5ca0b5beb","role":"sequence","name":"basis_map_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=0 and 0<=k<=2n.","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: 1;\n 1: 5, 30, 27;\n 2: 25, 300, 945, 1080, 405;\n 3: 125, 2250, 12150, 29700, 36450, 21870, 5103;\n 4: 625, 15000, 114750, 432000, 921375, 1166400, 867510, 349920, 59049;\n 5: 3125, 93750, 928125, 4725000, 14428125, 28248750, 36358875, 30618000, 16238475, 4920750, 649539;\n 6: 15625, 562500, 6834375, 43875000, 174656250, 464737500, 857941875, 1115370000, 1018595250, 639697500, 263063295, 63772920, 6908733;\n 7: 78125, 3281250, 47250000, 366187500, 1798453125, 6059812500, 14623284375, 25861275000, 33842458125, 32722987500, 23074872975, 11532269700, 3868890480, 781218270, 71744535;\n 8: 390625, 18750000, 312187500, 2835000000, 16567031250, 67614750000, 201823650000, 452709000000, 774833596875, 1018595250000, 1027570698000, 788658444000, 452176574850, 187492384800, 53090955900, 9183300480, 731794257;\n 9: 1953125, 105468750, 1993359375, 20756250000, 140674218750, 674552812500, 2401120968750, 6534756000000, 13849020309375, 23099845781250, 30467844680625, 31758914160000, 26006198195250, 16522766410500, 7974405065250, 2823864897600, 691545572865, 104603532030, 7360989291;\n","recurrence_sentence":"The row polynomials satisfy (5+15*n*x+(18*n-9)*x^2)*P(n,x) = (25+(75*n+150)*x+(315*n+315)*x^2+(405*n+270)*x^3+(162*n+81)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((25 + (75*n + 150)*x + (315*n + 315)*x^2 + (405*n + 270)*x^3 + (162*n + 81)*x^4)*P[n - 1, x])/(5 + 15*n*x + (18*n - 9)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((25 + (75*n + 150)*x + (315*n + 315)*x^2 + (405*n + 270)*x^3 + (162*n + 81)*x^4)*v[i - 1])/(5 + 15*n*x + (18*n - 9)*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-b30e7a5b4198051a","steps":["basis A013610","\u03b8+1"],"edges":["e-707e884996971c57","e-2bbe1ec6977b2f20"],"text":"A272866 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A272866 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}