{"id":"n-f891f11e3691237f","revision":0,"first_index":0,"fingerprint":"f6-5254523500bc8159","name":"basis_map_A013610 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["38","54"],["1444","4104","2916"],["54872","233928","332424","157464"],["2085136","11852352","25264224","23934528","8503056"],["79235168","562986720","1600067520","2273780160","1615580640","459165024"],["3010936384","25672194432","91203848640","172807292160","184176192960","104689625472","24794911296"],["114415582592","1138133953152","4852044747648","11491684928640","16330289109120","13923720187776","6595446404736","1338925209984"],["4347792138496","49427531679744","245836933880832","698694443661312","1241101972293120","1410936979027968","1002507853519872","407033263835136","72301961339136"],["165216101262848","2113026979309056","12010890198177792","39825583288694784","84891374904849408","120635111706891264","114285895301265408","69602688115808256","24727270777984512","3904305912313344"],["6278211847988224","89216694681937920","570517284413445120","2161960235672002560","5376453743973795840","9168268489723736064","10857160053620213760","8816340494669045760","4698181447817057280","1483636246679070720","210832519264920576"],["238572050223552512","3729257837705005056","26497358320535562240","112962422313862133760","321051094997292380160","638722704784086945792","907658580482649870336","921307581692915281920","654613281729176647680","310079975555925780480","88127993052736800768","11384956040305711104"],["9065737908494995456","154594688544862027776","1208279539416421638144","5723429397235681443840","18299912414845665669120","41608221911649092468736","68982052116681390145536","84023251450393873711104","74625914117126137835520","47132156284500718632960","20093182416023990575104","5191539954379404263424","614787626176508399616"],["344498040522809827328","6364148011763486810112","54262735679246571749376","282737412223442663325696","1004461859214862093393920","2569307703044331459944448","4868161963662943818842112","6917914369415762268880896","7373040314772062418149376","5820821301135838751170560","3308677371171950448033792","1282310368731712853065728","303705087331195149410304","33198531813531453579264"],["13090925539866773438464","260440518635244229459968","2405647948446598014222336","13674209391170136080842752","53437370910230663368556544","151874633113287148521160704","323732770583585763953000448","525761492075597932434948096","653742907909789534409244672","619335386440853243124547584","440054090365869409588494336","227396372055090412610322432","80785553230097909743140864","17661618924798733304168448","1792720717930698493280256"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","92","8464","778688","71639296","6590815232","606355001344","55784660123648","5132188731375616","472161363286556672","43438845422363213824","3996373778857415671808","367666387654882241806336","33825307664249166246182912","3111928305110923294648827904"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["38","54"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203870,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-f54c6d8207989193"]},{"id":203871,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-f54c6d8207989193"]},{"id":203872,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-f54c6d8207989193"]}],"region_exact":{},"incoming_edges":[{"id":"e-f54c6d8207989193","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-f891f11e3691237f","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"2e65c93a86b11691c58aaeb87cf2a9136f1ce892a93e54b1fcff91d4008b5322","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-76276d62adb9bc90","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","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":[{"id":"l-66213b984362bd72","kind":"fingerprint_related","node_a_id":"n-e7cce546ab56eb06","node_b_id":"n-f891f11e3691237f","fingerprint":"f6-5254523500bc8159","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5254523500bc8159"},"other_id":"n-e7cce546ab56eb06","other":{"id":"n-e7cce546ab56eb06","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","first_index":0,"fingerprint":"f6-5254523500bc8159","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["19","27"],["361","1026","729"]],"card_rows":[["1"],["19","27"],["361","1026","729"],["6859","29241","41553","19683"]],"row_sums_preview":["1","46","2116","97336","4477456","205962976","9474296896","435817657216"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2f02b2035f8f4541","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-c22af348c5e552a2","e-de9d9bd90fd19532"],"text":"A038322 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A038322 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}}],"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:              38,               54;\n  2:            1444,             4104,              2916;\n  3:           54872,           233928,            332424,            157464;\n  4:         2085136,         11852352,          25264224,          23934528,           8503056;\n  5:        79235168,        562986720,        1600067520,        2273780160,        1615580640,          459165024;\n  6:      3010936384,      25672194432,       91203848640,      172807292160,      184176192960,       104689625472,        24794911296;\n  7:    114415582592,    1138133953152,     4852044747648,    11491684928640,    16330289109120,     13923720187776,      6595446404736,     1338925209984;\n  8:   4347792138496,   49427531679744,   245836933880832,   698694443661312,  1241101972293120,   1410936979027968,   1002507853519872,   407033263835136,    72301961339136;\n  9: 165216101262848, 2113026979309056, 12010890198177792, 39825583288694784, 84891374904849408, 120635111706891264, 114285895301265408, 69602688115808256, 24727270777984512, 3904305912313344;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (38+54*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(38 + 54*x)*P[n - 1, x]];\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] = (38 + 54*x)*v[i - 1]); 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-4e096b5929fd3f2e","steps":["basis A013609","basis A013610"],"edges":["e-66940ce826511afb","e-f54c6d8207989193"],"text":"A038323 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A038323 \u2192 basis A013609 \u2192 basis A013610","chain":null}