{"id":"n-43ea2eaea8001298","revision":0,"first_index":0,"fingerprint":"f6-befc9532e898cdfb","name":"basis_map_A013609 of basis_map_A007318 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-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"],["13","16","4"],["169","416","360","128","16"],["2197","8112","12012","9088","3696","768","64"],["28561","140608","294736","342784","241504","105472","27904","4096","256"],["371293","2284880","6195540","9734400","9803040","6606336","3016320","921600","180480","20480","1024"],["4826809","35644128","118585272","234815360","307972080","281668608","184125696","86667264","29157120","6840320","1062912","98304","4096"],["62748517","540602608","2131221820","5092540544","8235023888","9528506624","8132220096","5199360000","2502221568","902107136","239891456","45645824","5877760","458752","16384"],["815730721","8031810176","36606519456","102464986624","197060846528","276001130496","291115062784","235821228032","148247520768","72560377856","27561189376","8040087552","1766309888","282591232","31064064","2097152","65536"],["10604499373","117465223824","607656638628","1949876378112","4347656868672","7150848187392","8988757290240","8827521490944","6864817669632","4258530156544","2112251590656","835741679616","261848186880","64094994432","11990532096","1654652928","158662656","9437184","262144"],["137858491849","1696719899680","9821397880840","35540760028800","90148187897040","170323041847296","248658043246080","287181288775680","266438002721280","200508060712960","123055023861760","61694787911680","25224899665920","8365772636160","2228789575680","469738979328","76499189760","9279897600","789053440","41943040","1048576"],["1792160394037","24263094565424","155377124813196","625959126066560","1779864194645680","3798733590482688","6318315983343936","8393177613410304","9053226828771840","8018337987911680","5873596292495360","3572944867491840","1807260397690880","759132590571520","263726225326080","75230330486784","17425471438848","3223581622272","464732815360","50289704960","3840933888","184549376","4194304"],["23298085122481","344094795655104","2415280777194480","10720555014138112","33775089046711584","80365200294872064","150037302009776896","225399299069767680","277256054521974528","282662733556842496","240863066924113920","172499175908966400","104155888219832320","53076669510451200","22803603969146880","8234144263831552","2485121317797888","621635426058240","127320718311424","20983817699328","2713498681344","265012903936","18371051520","805306368","16777216"],["302875106592253","4846001705476048","36997359174499828","179388086801527552","620266160942238304","1628031248637274624","3371428487031899008","5652248521242898432","7810866701941059328","9012309604869615616","8762847825010859008","7226949291829755904","5077465629097738240","3046487618789048320","1562297116645457920","684208216977965056","255267301229264896","80779778678849536","21541821426171904","4796462827307008","880299439292416","130796417449984","15333024858112","1364457422848","86587211776","3489660928","67108864"],["3937376385699289","67844023876664672","559713196982483544","2943386882033759616","11081658917771538096","31806217154566493184","72358134953380034304","133934211563217162240","205462957413247742208","264742886179333545984","289357442210819254272","270205144413439033344","216689633147590127616","149751586277140463616","89363526933406679040","46077411162197065728","20514994854209716224","7871246810405142528","2593589342318886912","730142274289139712","174354583652401152","34970995764756480","5813269759524864","786252014026752","84289051951104","6888590671872","403055837184","15032385536","268435456"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","33","1089","35937","1185921","39135393","1291467969","42618442977","1406408618241","46411484401953","1531578985264449","50542106513726817","1667889514952984961","55040353993448503713","1816331681783800622529"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["13","16","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200538,"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-ca1da524d3462c6a"]},{"id":200539,"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-ca1da524d3462c6a"]},{"id":200540,"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-ca1da524d3462c6a"]}],"region_exact":{},"incoming_edges":[{"id":"e-ca1da524d3462c6a","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-43ea2eaea8001298","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":"678177b50f455d940a35f5bbe026a23591b5d9b0b9fdcc7d708c0cba9b658d0f","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-351cbe3ca6567d98","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-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: 13, 16, 4;\n 2: 169, 416, 360, 128, 16;\n 3: 2197, 8112, 12012, 9088, 3696, 768, 64;\n 4: 28561, 140608, 294736, 342784, 241504, 105472, 27904, 4096, 256;\n 5: 371293, 2284880, 6195540, 9734400, 9803040, 6606336, 3016320, 921600, 180480, 20480, 1024;\n 6: 4826809, 35644128, 118585272, 234815360, 307972080, 281668608, 184125696, 86667264, 29157120, 6840320, 1062912, 98304, 4096;\n 7: 62748517, 540602608, 2131221820, 5092540544, 8235023888, 9528506624, 8132220096, 5199360000, 2502221568, 902107136, 239891456, 45645824, 5877760, 458752, 16384;\n 8: 815730721, 8031810176, 36606519456, 102464986624, 197060846528, 276001130496, 291115062784, 235821228032, 148247520768, 72560377856, 27561189376, 8040087552, 1766309888, 282591232, 31064064, 2097152, 65536;\n 9: 10604499373, 117465223824, 607656638628, 1949876378112, 4347656868672, 7150848187392, 8988757290240, 8827521490944, 6864817669632, 4258530156544, 2112251590656, 835741679616, 261848186880, 64094994432, 11990532096, 1654652928, 158662656, 9437184, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (13+16*x+4*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[(13 + 16*x + 4*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] = (13 + 16*x + 4*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-d3c3090977105807","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-33933faab90bc4be","e-ca1da524d3462c6a"],"text":"A272867 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A272867 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}