{"id":"n-6a3923af89150a19","revision":0,"first_index":0,"fingerprint":"f6-40d39f31cccb6263","name":"shift_x_plus_one of basis_map_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=0 and 0<=k<=2n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["29","33","9"],["841","1914","1611","594","81"],["24389","83259","117450","87615","36450","8019","729"],["707281","3219348","6373098","7166016","5005395","2223936","613818","96228","6561"],["20511149","116701365","297423855","447100830","438992865","294166323","136239165","43062030","8890155","1082565","59049"],["594823321","4061207502","12661037181","23831223570","30161935170","27041495382","17609360229","8392188222","2905013970","712327770","117448461","11691702","531441"],["17249876309","137404187151","506543335704","1145470578021","1775075832369","1994028238818","1674498210777","1067855804433","519671858517","192052660338","53057947509","10625808501","1458274104","122762871","4782969"],["500246412961","4553967345576","19379343798180","51171214525200","93836618234730","126713556596088","130339062484848","104172513433560","65380209340275","32329400720760","12553464995568","3787534657368","870463157130","147315445200","17314347780","1262703816","43046721"],["14507145975869","148573184649417","716784110267877","2164469272680924","4584326102322390","7231842473759442","8805909743844066","8462613960938016","6506770576539087","4032652150033155","2019342592719027","815067456404256","263213260210026","67085244012402","13197694191390","1933830894204","198746710857","12784876137","387420489"],["420707233300201","4787358172036770","25820218604982015","87760643208431490","210823999958230695","380486416569791004","535281119126441010","601097408677891620","547215795125184645","407833867025193510","250199391328797681","126569131145749710","52704493941902445","17967116770928820","4965465525849810","1095372200359404","188359321846095","24333880914090","2221856504415","127848761370","3486784401"],["12200509765705829","152716725687972963","910555524321393654","3440212090557250635","9242379192111767460","18781143868021435461","29976620201093968677","38520479531959529346","40523002617138747255","35295180060962729655","25639242116493180384","15597589520303806653","7957006174084090464","3399416866751463855","1211253799167417555","357330207101825826","86298892678045377","16779889970664381","2562685805407860","296033806952235","24316834412574","1265702737563","31381059609"],["353814783205469041","4831401867219508284","31555566740914876206","131188933459958015664","389750995278523070181","880613594327325210264","1572281146205438368986","2275352667875120236524","2716132482261533856906","2707503623921133583524","2272985986944321035046","1616081706481749912504","976226812266902866461","501542598563301696984","218920172345410230486","80928703179240902964","25195848914530326186","6550452131441196924","1404744291491160906","244172621716053624","33538476076059141","3503465177574384","261529750781406","12426899605164","282429536481"],["10260628712958602189","151786541995146218589","1077732030153624404715","4889295389593948943610","15916013767924017438015","39580277480823314554605","78164160810266152270935","125796027542103879928110","168005010341312394976440","188628151019219573328810","179709405551136522984780","146242439672423543870850","102098147752136819555415","61304955521479293018285","31685632061007978482715","14085181466666239064850","5371608374545021331580","1749784454604463601610","483665145021576100440","112391636831944637310","21673010946255932835","3405923796104091405","425044998317147715","40522048462505610","2772045900561015","121162271150349","2541865828329"],["297558232675799463481","4740410465386874211318","36355530418711560369873","178720802171250440687586","632610735397779440484000","1717060159791714238030530","3716153944276203953101215","6583724602787205373847490","9726891656079882862382805","12146545968699611580111000","12948346837689016641197310","11869094492860764190711680","9404231443962173421708105","6463264542995226437696610","3860830191747279421600875","2005840720239897859974810","905758319810863314100305","354773458743511299972480","120113651578478197750110","34968367345278578211000","8690427638089779987405","1825505882800966170090","319778888738935695615","45854938514815359930","5243021915233284000","459689656744424106","29020482162032193","1174342012687998","22876792454961"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","71","5041","357911","25411681","1804229351","128100283921","9095120158391","645753531245761","45848500718449031","3255243551009881201","231122292121701565271","16409682740640811134241","1165087474585497590531111","82721210695570328927708881"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["29","33","9"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191977,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-8a40327d880a2a66"]},{"id":191978,"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-394632dec17184f9","e-8a40327d880a2a66"]},{"id":191979,"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-394632dec17184f9","e-8a40327d880a2a66"]},{"id":191980,"position":3,"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-394632dec17184f9"]}],"region_exact":{},"incoming_edges":[{"id":"e-394632dec17184f9","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-6a3923af89150a19","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":"cfdcf598b109cb37d1e6e5bdf18c2f1756c642f43bd63e1c1a847cf922095e20","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-6d8c9ba5ca0b5beb","role":"sequence","name":"basis_map_A013610 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]"}]},{"id":"e-8a40327d880a2a66","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-6a3923af89150a19","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"19ce715eb191ba6750aa31a74d49b5beacb8d09ec482a3a2213118880375e544","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-6d8c9ba5ca0b5beb","role":"sequence","name":"basis_map_A013610 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: 29, 33, 9;\n 2: 841, 1914, 1611, 594, 81;\n 3: 24389, 83259, 117450, 87615, 36450, 8019, 729;\n 4: 707281, 3219348, 6373098, 7166016, 5005395, 2223936, 613818, 96228, 6561;\n 5: 20511149, 116701365, 297423855, 447100830, 438992865, 294166323, 136239165, 43062030, 8890155, 1082565, 59049;\n 6: 594823321, 4061207502, 12661037181, 23831223570, 30161935170, 27041495382, 17609360229, 8392188222, 2905013970, 712327770, 117448461, 11691702, 531441;\n 7: 17249876309, 137404187151, 506543335704, 1145470578021, 1775075832369, 1994028238818, 1674498210777, 1067855804433, 519671858517, 192052660338, 53057947509, 10625808501, 1458274104, 122762871, 4782969;\n 8: 500246412961, 4553967345576, 19379343798180, 51171214525200, 93836618234730, 126713556596088, 130339062484848, 104172513433560, 65380209340275, 32329400720760, 12553464995568, 3787534657368, 870463157130, 147315445200, 17314347780, 1262703816, 43046721;\n 9: 14507145975869, 148573184649417, 716784110267877, 2164469272680924, 4584326102322390, 7231842473759442, 8805909743844066, 8462613960938016, 6506770576539087, 4032652150033155, 2019342592719027, 815067456404256, 263213260210026, 67085244012402, 13197694191390, 1933830894204, 198746710857, 12784876137, 387420489;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (29+33*x+9*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(29 + 33*x + 9*x^2)*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] = (29 + 33*x + 9*x^2)*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-b30e7a5b4198051a","steps":["basis A013610","basis A007318"],"edges":["e-707e884996971c57","e-394632dec17184f9"],"text":"A272866 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A272866 \u2192 basis A013610 \u2192 basis A007318","chain":null}