{"id":"n-ed4eca858e43ef4e","revision":1,"first_index":0,"fingerprint":"f6-9a1b24eba84fcdcc","name":"obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["7","6"],["42","114","36"],["252","1512","1440","216"],["1512","17280","35100","16092","1296"],["9072","182088","676188","658368","167508","7776"],["54432","1825416","11344212","19917252","10867932","1666980","46656"],["326592","17711784","173965644","501027120","492454080","164923128","16099236","279936"],["1959552","168084072","2507692932","11136580164","17824837320","10818091728","2360481588","152311428","1679616"],["11757312","1570703400","34582427388","226743541056","553269998124","546581740968","218433142260","32390029872","1420351524","10077696"],["70543872","14519283048","461654136180","4327845368292","15368740468164","23044857266052","15048902268252","4143994653420","430834809708","13110688836","60466176"],["423263232","133182893736","6014729959020","78670590614640","392840887231080","851183993083536","843507876285648","382119830331120","74988250097400","5598414432360","120142748772","362797056"],["2539579392","1214971911144","76932041849316","1377256699453860","9418918418633520","28437069369328416","40605078073801752","27986402380273848","9115469178917040","1308283179315480","71466810310692","1095252425604","2176782336"],["15237476352","11040321143592","970208159878620","23411648056919616","214766326115501940","878611985093874096","1739836721554917336","1727668712850444000","860044501132233768","207028113636038760","22180296919453452","899956321479072","9947608297380","13060694016"],["91424858112","100042046381160","12102906737688660","388798798904976132","4704705106531658316","25511017717748047716","68031615687214445640","93503317519911542760","67182558959457875736","24865739542731315888","4521088640952539172","367588710145401300","11213990737353180","90109675560132","78364164096"],["548549148672","904727628537768","149706936644870988","6337607482262014128","99781518224289642768","704483579660537067912","2472069350406470441004","4560437862698413485024","4534966160865716387904","2430397809164760612984","684322545224337410340","95649228382242079728","5982121678854301488","138592987953898968","814709377835748","470184984576"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["13","192","3420","71280","1701000","45722880","1366787520","44971718400","1615013769600","62854589952000","2635178683737600","118399191092582400","5675439486068352000","289099937902583808000","15594717804168222720000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["7","6"],["42","114","36"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","0"],["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","6"],["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162880,"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-0e437c53d6b0d63a"]}],"region_exact":{},"incoming_edges":[{"id":"e-0e437c53d6b0d63a","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-ed4eca858e43ef4e","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":"f9e7cbe7a067482137edd8046acf30f1b8358df34e37a1dcd6dcfd36f70f31fd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-61cf3a32da4a0e22","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.","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-61cf3a32da4a0e22","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.","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:        7,           6;\n  1:       42,         114,           36;\n  2:      252,        1512,         1440,           216;\n  3:     1512,       17280,        35100,         16092,           1296;\n  4:     9072,      182088,       676188,        658368,         167508,           7776;\n  5:    54432,     1825416,     11344212,      19917252,       10867932,        1666980,          46656;\n  6:   326592,    17711784,    173965644,     501027120,      492454080,      164923128,       16099236,        279936;\n  7:  1959552,   168084072,   2507692932,   11136580164,    17824837320,    10818091728,     2360481588,     152311428,      1679616;\n  8: 11757312,  1570703400,  34582427388,  226743541056,   553269998124,   546581740968,   218433142260,   32390029872,   1420351524,    10077696;\n  9: 70543872, 14519283048, 461654136180, 4327845368292, 15368740468164, 23044857266052, 15048902268252, 4143994653420, 430834809708, 13110688836, 60466176;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+3*n*x)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (5+(3*n+6)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 7 + 6*x, P(1,x) = 42 + 114*x + 36*x^2.","mathematica":"P[0, x_] := 7 + 6*x;\nP[1, x_] := 42 + 114*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[(6 + 3*n*x)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (5 + (3*n + 6)*x)*P[n - 1, x] + (3*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] = 7 + 6*x; v[2] = 42 + 114*x + 36*x^2; for(n = 2, N, my(i = n + 1); v[i] = (6 + 3*n*x)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (5 + (3*n + 6)*x)*v[i - 1] + (3*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-61cf3a32da4a0e22","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-0e437c53d6b0d63a"],"text":"A257626 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257626 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}