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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":"1030bcd685289e3eaa8070c920f1e45d54af2f8c6765e2e87ba83a2bb0e3487d","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-feae95e8ddee939b","role":"sequence","name":"basis_map_A013609 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, 40, 36;\n 2: 25, 400, 1260, 1280, 400;\n 3: 125, 3000, 16200, 35200, 36000, 17280, 3136;\n 4: 625, 20000, 153000, 512000, 910000, 921600, 533120, 163840, 20736;\n 5: 3125, 125000, 1237500, 5600000, 14250000, 22320000, 22344000, 14336000, 5702400, 1280000, 123904;\n 6: 15625, 750000, 9112500, 52000000, 172500000, 367200000, 527240000, 522240000, 357696000, 166400000, 50181120, 8847360, 692224;\n 7: 78125, 4375000, 63000000, 434000000, 1776250000, 4788000000, 8986600000, 12108800000, 11884320000, 8512000000, 4401689600, 1599897600, 387645440, 56197120, 3686400;\n 8: 390625, 25000000, 416250000, 3360000000, 16362500000, 53424000000, 124028800000, 211968000000, 272095200000, 264960000000, 196016128000, 109412352000, 45306060800, 13487308800, 2727936000, 335544320, 18939904;\n 9: 1953125, 140625000, 2657812500, 24600000000, 138937500000, 532980000000, 1475586000000, 3059712000000, 4863304800000, 6008800000000, 5811949440000, 4405985280000, 2605704192000, 1188569088000, 409743360000, 103179878400, 17898209280, 1911029760, 94633984;\n","recurrence_sentence":"The row polynomials satisfy (25+(150*n-50)*x+(100*n^2+160*n-120)*x^2+(160*n^2-40*n-40)*x^3+(64*n^2-64*n+16)*x^4)*P(n,x) = (125+(750*n+750)*x+(500*n^2+3300*n+1800)*x^2+(1800*n^2+5600*n+2200)*x^3+(2320*n^2+4560*n+1440)*x^4+(1280*n^2+1760*n+480)*x^5+(256*n^2+256*n+64)*x^6)*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[((125 + (750*n + 750)*x + (500*n^2 + 3300*n + 1800)*x^2 + (1800*n^2 + 5600*n + 2200)*x^3 + (2320*n^2 + 4560*n + 1440)*x^4 + (1280*n^2 + 1760*n + 480)*x^5 + (256*n^2 + 256*n + 64)*x^6)*P[n - 1, x])/(25 + (150*n - 50)*x + (100*n^2 + 160*n - 120)*x^2 + (160*n^2 - 40*n - 40)*x^3 + (64*n^2 - 64*n + 16)*x^4)]];\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] = ((125 + (750*n + 750)*x + (500*n^2 + 3300*n + 1800)*x^2 + (1800*n^2 + 5600*n + 2200)*x^3 + (2320*n^2 + 4560*n + 1440)*x^4 + (1280*n^2 + 1760*n + 480)*x^5 + (256*n^2 + 256*n + 64)*x^6)*v[i - 1])/(25 + (150*n - 50)*x + (100*n^2 + 160*n - 120)*x^2 + (160*n^2 - 40*n - 40)*x^3 + (64*n^2 - 64*n + 16)*x^4)); 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 A013609","(\u03b8+1)\u00b2"],"edges":["e-89275777750acd05","e-4acd918eedfa0d2e"],"text":"A272866 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A272866 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}