{"id":"n-9aad4442f558f61d","revision":1,"first_index":0,"fingerprint":"f6-435a2c1b6d6be7b5","name":"obreschkoff_combination of Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2","45","108","27"],["2","111","693","729","81"],["2","249","3312","8856","4698","243"],["2","531","13437","73386","101088","29403","729"],["2","1101","49356","486945","1361610","1072359","180792","2187"],["2","2247","170109","2804031","13916529","22484061","10829295","1100061","6561"],["2","4545","562032","14672502","118790064","339856884","342484200","105723954","6652854","19683"],["2","9147","1804293","71784360","894925908","4139408286","7430920074","4923654336","1008097650","40094271","59049"],["2","18357","5678172","334561671","6160583628","43311477762","125570541600","149857530894","67842852642","9453882681","241156116","177147"],["2","36783","17621973","1504470645","39657600729","405356511942","1777071847410","3437832740634","2843991552420","905648704443","87617437641","1448885313","531441"],["2","73641","54153360","6586494372","242596920510","3486778902153","22049587761984","64465756351200","87041797795278","51515836126875","11803334843088","805220271324","8699689170","1594323"],["2","147363","165258477","28255545846","1426650707376","28095723615213","247392546612687","1039940355924468","2127278680818714","2073579271130349","899885520488043","151040532351222","7355732276628","52218861219","4782969"],["2","294813","501817356","119344756509","8134381913670","215028881686161","2565256796118072","14937948643825089","43920557783216046","65155851065564055","47071308802023180","15275621320027767","1905699593121354","66906557073531","313380128880","14348907"],["2","589719","1518423885","498082087035","45257659708329","1579356020235483","24977826236251305","195780288356418015","795673331250788025","1699522855353301941","1879988390189400663","1027926505590793857","253457704228629915","23783045817128985","606703063794291","1880496006885","43046721"],["2","1179537","4582988496","2059525238850","246918407291460","11220436723070322","231131438169396552","2383495806937007826","12992591088199591200","38352775866054171660","61279022146375661184","51676324843346827254","21752251101928759212","4129133147189247150","294296532743042280","5489475384221046","11283664788846","129140163"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["182","1616","17360","218576","3154352","51312896","928766720","18510757376","402778460672","9500147941376","241426250977280","6576075658520576","191122747065915392","5903343643906605056","193107992149361623040"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","45","108","27"],["2","111","693","729","81"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"],["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","-3"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","9"],["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162862,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-4796775b805410d0"]}],"region_exact":{},"incoming_edges":[{"id":"e-4796775b805410d0","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-9aad4442f558f61d","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":"868815df6b6ee6692e8f76ab3876a1f0276ba3daec16a5907f35682d7e793b6b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-02156bdfcfec6826","role":"sequence","name":"Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-02156bdfcfec6826","role":"sequence","name":"Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"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: 2,    45,      108,         27;\n  1: 2,   111,      693,        729,          81;\n  2: 2,   249,     3312,       8856,        4698,          243;\n  3: 2,   531,    13437,      73386,      101088,        29403,           729;\n  4: 2,  1101,    49356,     486945,     1361610,      1072359,        180792,          2187;\n  5: 2,  2247,   170109,    2804031,    13916529,     22484061,      10829295,       1100061,          6561;\n  6: 2,  4545,   562032,   14672502,   118790064,    339856884,     342484200,     105723954,       6652854,        19683;\n  7: 2,  9147,  1804293,   71784360,   894925908,   4139408286,    7430920074,    4923654336,    1008097650,     40094271,       59049;\n  8: 2, 18357,  5678172,  334561671,  6160583628,  43311477762,  125570541600,  149857530894,   67842852642,   9453882681,   241156116,     177147;\n  9: 2, 36783, 17621973, 1504470645, 39657600729, 405356511942, 1777071847410, 3437832740634, 2843991552420, 905648704443, 87617437641, 1448885313, 531441;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(3*n+3)*x)*P(n-2,x) + (x-3*x^2)*d/dx P(n-2,x) + (3*n+9)*x*P(n-1,x) + (x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 45*x + 108*x^2 + 27*x^3, P(1,x) = 2 + 111*x + 693*x^2 + 729*x^3 + 81*x^4.","mathematica":"P[0, x_] := 2 + 45*x + 108*x^2 + 27*x^3;\nP[1, x_] := 2 + 111*x + 693*x^2 + 729*x^3 + 81*x^4;\nP[n_, x_] := P[n, x] = Expand[(1 + (3*n + 3)*x)*P[n - 2, x] + (x - 3*x^2)*D[P[n - 2, x], x] + (3*n + 9)*x*P[n - 1, x] + (x - 3*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + 45*x + 108*x^2 + 27*x^3; v[2] = 2 + 111*x + 693*x^2 + 729*x^3 + 81*x^4; for(n = 2, N, my(i = n + 1); v[i] = (1 + (3*n + 3)*x)*v[i - 2] + (x - 3*x^2)*deriv(v[i - 2], x) + (3*n + 9)*x*v[i - 1] + (x - 3*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-02156bdfcfec6826","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-4796775b805410d0"],"text":"A156366 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A156366 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}