{"id":"n-4cf7a5d85123554f","revision":0,"first_index":0,"fingerprint":"f6-421bc4a701fbf074","name":"reversal of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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":[["81","234","169"],["729","3159","4563","2197"],["6561","37908","82134","79092","28561"],["59049","426465","1232010","1779570","1285245","371293"],["531441","4605822","16632135","32032260","34701615","20049822","4826809"],["4782969","48361131","209564901","504508095","728733915","631569393","304088967","62748517"],["43046721","497428776","2514778812","7264916568","13117210470","15157665432","10947202812","4517893224","815730721"],["387420489","5036466357","29099583396","98076373668","212498809614","306942724998","295574475924","182974675572","66074188401","10604499373"],["3486784401","50364663570","327370313205","1260981947160","3187482144210","5524969049964","6650425708290","5489240267160","2973338478045","954404943570","137858491849"],["31381059609","498610169343","3601073445255","15604651596105","45080104610970","91161989324406","131678429024142","135858696612210","98120169775485","47243044706715","13647990693051","1792160394037"],["282429536481","4895445299004","38891593208754","187255819153260","608581412248095","1406499263862264","2370211722434556","2934547846823736","2649244583938095","1700749609441740","736991497424754","193553322555996","23298085122481"],["2541865828329","47730591665289","413665127765838","2190893084093142","7911558359225235","20570051733985611","39616395932120436","57223683013062852","61992323264151423","49746926076170895","28742668399565406","11322869369525766","2725875959330277","302875106592253"],["22876792454961","462619580755878","4343483841541299","25095684417794172","99685635326237961","287980724275798554","623958235930896867","1030026294235131336","1301838788547179883","1253622537119506554","905394054586310289","475560513520082172","171730185437807451","38162263430623878","3937376385699289"],["205891132094649","4460974528717395","45105409123698105","282326449700184435","1223414615367465885","3887739777723280479","9359373538963453005","17381693715217841295","25106890921981326315","28206507085188897465","24445639473830377803","16050167331302773305","7727858344701335295","2575952781567111765","531545812069404015","51185893014090757"],["1853020188851841","42825355475686992","463941350986609080","3127308365909735280","14680975384409590620","50894047999286580816","134774978961073723272","278107099443485460720","451924036595663873670","580248145752457319280","586695347371929067272","462244819141519871184","278202900409248070620","123645733515221364720","38271298468997089080","7370768594029069008","665416609183179841"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["484","10648","234256","5153632","113379904","2494357888","54875873536","1207269217792","26559922791424","584318301411328","12855002631049216","282810057883082752","6221821273427820544","136880068015412051968","3011361496339065143296"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["81","234","169"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["9","13"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193612,"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-974d9a6ffed76db8"]},{"id":193613,"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-974d9a6ffed76db8"]},{"id":193614,"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-974d9a6ffed76db8"]}],"region_exact":{},"incoming_edges":[{"id":"e-974d9a6ffed76db8","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-4cf7a5d85123554f","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"106653b94a81a839846ed48d3ca64df50c04a48ae8ad96d8cfa9c04a2898faac","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-7e380db7ce29966b","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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-1117c721eb5be645","kind":"fingerprint_related","node_a_id":"n-7e380db7ce29966b","node_b_id":"n-4cf7a5d85123554f","fingerprint":"f6-421bc4a701fbf074","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-421bc4a701fbf074"},"other_id":"n-7e380db7ce29966b","other":{"id":"n-7e380db7ce29966b","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-421bc4a701fbf074","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["169","234","81"],["2197","4563","3159","729"],["28561","79092","82134","37908","6561"]],"card_rows":[["169","234","81"],["2197","4563","3159","729"],["28561","79092","82134","37908","6561"],["371293","1285245","1779570","1232010","426465","59049"]],"row_sums_preview":["484","10648","234256","5153632","113379904","2494357888","54875873536","1207269217792"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["basis A013610"],"edges":["e-c80be26a429cc3fc"],"text":"A038305 \u2192 basis A013610"},"headline":"A038305 \u2192 basis A013610","chain":null}},{"id":"l-8a81d0c5deb9587e","kind":"fingerprint_related","node_a_id":"n-38deee6c8695ee46","node_b_id":"n-4cf7a5d85123554f","fingerprint":"f6-421bc4a701fbf074","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-421bc4a701fbf074"},"other_id":"n-38deee6c8695ee46","other":{"id":"n-38deee6c8695ee46","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-421bc4a701fbf074","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","13"],["81","234","169"]],"card_rows":[["1"],["9","13"],["81","234","169"],["729","3159","4563","2197"]],"row_sums_preview":["1","22","484","10648","234256","5153632","113379904","2494357888"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-366b2525683554ec","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-b30dba0e9f8b339c","e-0a277ec2fb66d424"],"text":"A038279 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A038279 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-d3629059ab1e08fb","kind":"fingerprint_related","node_a_id":"n-c780c8343b508644","node_b_id":"n-4cf7a5d85123554f","fingerprint":"f6-421bc4a701fbf074","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-421bc4a701fbf074"},"other_id":"n-c780c8343b508644","other":{"id":"n-c780c8343b508644","name":"shift_x_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-421bc4a701fbf074","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["13","9"],["169","234","81"]],"card_rows":[["1"],["13","9"],["169","234","81"],["2197","4563","3159","729"]],"row_sums_preview":["1","22","484","10648","234256","5153632","113379904","2494357888"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-39440d1ff583ab52","e-906d296b8243e51b"],"text":"A013610 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A013610 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-f9e3e3befad0b0a7","kind":"fingerprint_related","node_a_id":"n-ef779be944ea4a4c","node_b_id":"n-4cf7a5d85123554f","fingerprint":"f6-421bc4a701fbf074","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-421bc4a701fbf074"},"other_id":"n-ef779be944ea4a4c","other":{"id":"n-ef779be944ea4a4c","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-421bc4a701fbf074","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["234","162"],["4563","6318","2187"],["79092","164268","113724","26244"]],"card_rows":[["234","162"],["4563","6318","2187"],["79092","164268","113724","26244"],["1285245","3559140","3696030","1705860","295245"]],"row_sums_preview":["396","13068","383328","10541520","278296128","7142933952","179593767936","4444945756416"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["basis A013610","D"],"edges":["e-c80be26a429cc3fc","e-2f6fa3fef5285de9"],"text":"A038305 \u2192 basis A013610 \u2192 D"},"headline":"A038305 \u2192 basis A013610 \u2192 D","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:          81,          234,           169;\n  1:         729,         3159,          4563,           2197;\n  2:        6561,        37908,         82134,          79092,          28561;\n  3:       59049,       426465,       1232010,        1779570,        1285245,         371293;\n  4:      531441,      4605822,      16632135,       32032260,       34701615,       20049822,         4826809;\n  5:     4782969,     48361131,     209564901,      504508095,      728733915,      631569393,       304088967,        62748517;\n  6:    43046721,    497428776,    2514778812,     7264916568,    13117210470,    15157665432,     10947202812,      4517893224,      815730721;\n  7:   387420489,   5036466357,   29099583396,    98076373668,   212498809614,   306942724998,    295574475924,    182974675572,    66074188401,    10604499373;\n  8:  3486784401,  50364663570,  327370313205,  1260981947160,  3187482144210,  5524969049964,   6650425708290,   5489240267160,  2973338478045,   954404943570,   137858491849;\n  9: 31381059609, 498610169343, 3601073445255, 15604651596105, 45080104610970, 91161989324406, 131678429024142, 135858696612210, 98120169775485, 47243044706715, 13647990693051, 1792160394037;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (9+13*x)*P(n-1,x) for n >= 1, with P(0,x) = 81 + 234*x + 169*x^2.","mathematica":"P[0, x_] := 81 + 234*x + 169*x^2;\nP[n_, x_] := P[n, x] = Expand[(9 + 13*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] = 81 + 234*x + 169*x^2; for(n = 1, N, my(i = n + 1); v[i] = (9 + 13*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-4397e742863fdbe2","steps":["basis A013610","reverse"],"edges":["e-c80be26a429cc3fc","e-974d9a6ffed76db8"],"text":"A038305 \u2192 basis A013610 \u2192 reverse"},"headline":"A038305 \u2192 basis A013610 \u2192 reverse","chain":null}
