{"id":"n-23763ec4ded1b991","revision":0,"first_index":0,"fingerprint":"f6-b21306f08ea8390f","name":"basis_map_A013609 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1521","3276","1764"],["59319","191646","206388","74088"],["2313441","9965592","16098264","11557728","3111696"],["90224199","485822610","1046387160","1126878480","606780720","130691232"],["3518743761","22736498148","61213648860","87896521440","70993344240","30581748288","5489031744"],["137231006679","1034510665734","3342265227756","5998937588280","6460394325840","4174408641312","1498505666112","230539333248"],["5352009260481","46109618244144","173797791843312","374333705508672","503910757415520","434138498696448","233766883913472","71928271973376","9682651996416"],["208728361158759","2023059500461818","8714717848143216","21898521772257312","35374535170569504","38095653260613312","27350725417876224","12623411731327488","3398610850742016","406671383849472"],["8140406085191601","87665911686678780","424842495096981780","1220060498740050240","2299344786087017760","2971460954327838336","2666695728242931840","1641043525072573440","662729115894693120","158601839701294080","17080198121677824"],["317475837322472439","3760867611358519662","20250825599622798180","65425744244935194120","140916987604475802720","212459458234440441024","228802493483243551872","176001918064033501440","94770263572941116160","34020094615927580160","7327404994199786496","717368321110468608"],["12381557655576425121","160007822010526109256","947738638062346954824","3402138700736630094240","8243643774861834459120","14204432350531160914176","17846594491692997046016","16473779530793535734784","11088120838034110590720","5307134760084702504960","1714612768642750040064","335728374279699308544","30129469486639681536"],["482880748567480579719","6760330479944728116066","43682135408873627826888","172488432127347145777968","464391932650550007863760","900205900214912322935904","1292603343898335643190016","1392034370352053769589248","1124335452976658813899008","672679330840736042503680","289769557900624756770816","85107142879903774715904","15275641029726318538752","1265437718438866624512"],["18832349194131742609041","283933880157678580874772","1987537161103750066123404","8561698540139231054070048","25355799522720030429361296","54612491279704680924778176","88220178221061407647718592","108578680887460194027961344","102314526220875952064809728","73456582927808375841401856","39553544653435279299216384","15489500004142486998294528","4170250001115284961079296","690928994267621176983552","53148384174432398229504"],["734461618571137961752599","11864379992302997843695830","89439172249668752975553180","417382803831787513885914840","1348467520071928891016032560","3194830739862723834099523296","5734311584368991497101708480","7939816039895526688294673280","8550571119887490279701955840","7162016835461316644536680960","4627764724451927678008316928","2265339375605838723500574720","813198750217480567410462720","202096730823279194267688960","31091804742042952964259840","2232232135326160725639168"],["28644003124274380508351361","493558207679804710297746528","3986431677413807275481798880","20034374583925800666523912320","70120311043740302332833693120","181234034697667242952554776064","357821042864625069419146609152","550493912099423183721764014080","666944547351224241816752555520","638442643618265940884412702720","481287531343000478512864960512","282714354075608672692871725056","126859005033926968516032184320","42036120011242072407679303680","9700643079517401324849070080","1392912852443524292798840832","93753749683698750476845056"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["6561","531441","43046721","3486784401","282429536481","22876792454961","1853020188851841","150094635296999121","12157665459056928801","984770902183611232881","79766443076872509863361","6461081889226673298932241","523347633027360537213511521","42391158275216203514294433201","3433683820292512484657849089281"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1521","3276","1764"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["39","42"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201198,"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-cdf037e7e0724686"]},{"id":201199,"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-cdf037e7e0724686"]},{"id":201200,"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-cdf037e7e0724686"]}],"region_exact":{},"incoming_edges":[{"id":"e-cdf037e7e0724686","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-23763ec4ded1b991","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":"bce8059ab847942777ef5529bfaa8cf7f38ca371bd838b9020e58b463d29cd22","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-63d702ebcae1deea","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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-e73940c8d795c296","kind":"fingerprint_related","node_a_id":"n-0b1d1ff3e2f24ef7","node_b_id":"n-23763ec4ded1b991","fingerprint":"f6-b21306f08ea8390f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b21306f08ea8390f"},"other_id":"n-0b1d1ff3e2f24ef7","other":{"id":"n-0b1d1ff3e2f24ef7","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","first_index":0,"fingerprint":"f6-b21306f08ea8390f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["13","14"],["169","364","196"]],"card_rows":[["1"],["13","14"],["169","364","196"],["2197","7098","7644","2744"]],"row_sums_preview":["1","27","729","19683","531441","14348907","387420489","10460353203"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-81e6fb4f9f226d77","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-4bd827707ee8c2a0","e-d599796f1b627386"],"text":"A038327 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A038327 \u2192 x \u21a6 x+1 \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:               1521,                3276,                 1764;\n  1:              59319,              191646,               206388,                74088;\n  2:            2313441,             9965592,             16098264,             11557728,               3111696;\n  3:           90224199,           485822610,           1046387160,           1126878480,             606780720,             130691232;\n  4:         3518743761,         22736498148,          61213648860,          87896521440,           70993344240,           30581748288,            5489031744;\n  5:       137231006679,       1034510665734,        3342265227756,        5998937588280,         6460394325840,         4174408641312,         1498505666112,          230539333248;\n  6:      5352009260481,      46109618244144,      173797791843312,      374333705508672,       503910757415520,       434138498696448,       233766883913472,        71928271973376,        9682651996416;\n  7:    208728361158759,    2023059500461818,     8714717848143216,    21898521772257312,     35374535170569504,     38095653260613312,     27350725417876224,     12623411731327488,     3398610850742016,      406671383849472;\n  8:   8140406085191601,   87665911686678780,   424842495096981780,  1220060498740050240,   2299344786087017760,   2971460954327838336,   2666695728242931840,   1641043525072573440,   662729115894693120,   158601839701294080,   17080198121677824;\n  9: 317475837322472439, 3760867611358519662, 20250825599622798180, 65425744244935194120, 140916987604475802720, 212459458234440441024, 228802493483243551872, 176001918064033501440, 94770263572941116160, 34020094615927580160, 7327404994199786496, 717368321110468608;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (39+42*x)*P(n-1,x) for n >= 1, with P(0,x) = 1521 + 3276*x + 1764*x^2.","mathematica":"P[0, x_] := 1521 + 3276*x + 1764*x^2;\nP[n_, x_] := P[n, x] = Expand[(39 + 42*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] = 1521 + 3276*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = (39 + 42*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-7dcf11eddf2b9a8a","steps":["basis A013610","basis A013609"],"edges":["e-632cc4d34c6e419b","e-cdf037e7e0724686"],"text":"A038321 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A038321 \u2192 basis A013610 \u2192 basis A013609","chain":null}
