{"id":"n-3cd27d08ccf7c44d","revision":1,"first_index":1,"fingerprint":"f6-f40ca4b4bfa77e79","name":"obreschkoff_combination of Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["2","-1"],["2","-4","2"],["2","-10","14","-6"],["2","-22","62","-66","24"],["2","-46","230","-450","384","-120"],["2","-94","782","-2490","3720","-2640","720"],["2","-190","2534","-12306","28560","-34440","20880","-5040"],["2","-382","7982","-56826","192024","-349440","352800","-186480","40320"],["2","-766","24710","-251250","1187424","-3056760","4566240","-3961440","1854720","-362880"],["2","-1534","75662","-1079130","6942120","-24277680","50304240","-63655200","48384000","-20321280","3628800"],["2","-3070","230054","-4543506","39027120","-180376680","497795760","-861371280","944697600","-638668800","243129600","-39916800"],["2","-6142","696302","-18864186","213309624","-1277395680","4566830400","-10375540560","15393248640","-14888966400","9061113600","-3153427200","479001600"],["2","-12286","2101190","-77545650","1142004864","-8730922200","39632186880","-114972137280","221543562240","-287428901760","248561913600","-137513376000","44068147200","-6227020800"],["2","-24574","6328142","-316486170","6020206920","-58095557520","329810841360","-1197202406400","2913669158400","-4868181077760","5608470067200","-4384341561600","2223046425600","-660064204800","87178291200"],["2","-49150","19033574","-1284929106","31366979280","-378674379720","2657249234640","-11886295140720","35800641676800","-74904833203200","110374981516800","-114305269478400","81511702272000","-38140502400000","10548573235200","-1307674368000"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["1","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],"nonn":false,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","-1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162751,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-aa85cc5c316d6389"]}],"region_exact":{},"incoming_edges":[{"id":"e-aa85cc5c316d6389","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-3cd27d08ccf7c44d","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"155586ae7b6126cd422c97e6f5c6aab2c8e3ec0643d5ba89b97454ec89ec720a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-2ae6a2215aff79d4","role":"sequence","name":"Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-2ae6a2215aff79d4","role":"sequence","name":"Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 2,    -1;\n   2: 2,    -4,     2;\n   3: 2,   -10,    14,       -6;\n   4: 2,   -22,    62,      -66,      24;\n   5: 2,   -46,   230,     -450,     384,      -120;\n   6: 2,   -94,   782,    -2490,    3720,     -2640,      720;\n   7: 2,  -190,  2534,   -12306,   28560,    -34440,    20880,     -5040;\n   8: 2,  -382,  7982,   -56826,  192024,   -349440,   352800,   -186480,    40320;\n   9: 2,  -766, 24710,  -251250, 1187424,  -3056760,  4566240,  -3961440,  1854720,   -362880;\n  10: 2, -1534, 75662, -1079130, 6942120, -24277680, 50304240, -63655200, 48384000, -20321280, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1-x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2 - x.","mathematica":"P[1, x_] := 2 - x;\nP[n_, x_] := P[n, x] = Expand[(1 - x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2 - x; for(n = 2, N, my(i = n); v[i] = (1 - x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2ae6a2215aff79d4","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-aa85cc5c316d6389"],"text":"A163626 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A163626 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}