{"id":"n-4182d3b7552bf009","revision":0,"first_index":0,"fingerprint":"f6-e01ea07d291a6f88","name":"basis_map_A013610 of basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","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":false,"rows":[["1"],["9","18"],["81","288","243"],["729","3690","5913","2916"],["6561","43056","101250","99144","32805"],["59049","476082","1475982","2169504","1476225","354294"],["531441","5081904","19561581","38415384","39963051","20312856","3720087"],["4782969","52911594","243314847","598471092","841113639","663828858","264126177","38263752"],["43046721","540774432","2892033396","8550680112","15165029790","16325867520","10222799076","3290682672","387420489"],["387420489","5448100626","33219793548","114752321568","245966575590","336358456740","289739507436","148539885264","39646030041","3874204890"],["3486784401","54263082240","371546842455","1468550846160","3692730844890","6132533422608","6749877844926","4796667423216","2060947861317","464904586800","38354628411"],["31381059609","535439329578","4067647269525","18105415729380","52260095857410","102190130194572","137190075202530","125236648075968","75102624054117","27538623199098","5331293349129","376572715308"],["282429536481","5242598270928","43757823068118","216596011375800","705866824167435","1589030142412608","2520306191666340","2817389216739024","2180977228864071","1123228077892272","356621334965478","60000585972408","3671583974253"],["2541865828329","50996183180850","463880584506618","2527628649169536","9179883705504015","23387609237008842","42819559008310476","56726609178003696","53971679485948887","36042313096608390","16165938910438746","4497282414685008","664556699339793","35586121596606"],["22876792454961","493280837310096","4856939635623345","28889200004808744","115702783039774125","329151642360965928","683506960176423789","1047853675093570704","1184883145774897923","975823012097612640","569897961084476739","225180692863585992","55437246427246047","7259568805707624","343151886824415"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","27","612","13248","282816","6011136","127586304","2706812928","57418334208","1217932296192","25833864167424","547965946036224","11622970048905216","246536065140129792","5229302136881283072"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["9","18"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-9","-54","-81"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["10","18"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203176,"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-de275f2bc42bbd59"]},{"id":203177,"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-de275f2bc42bbd59"]},{"id":203178,"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-de275f2bc42bbd59"]},{"id":203179,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-de275f2bc42bbd59"]}],"region_exact":{},"incoming_edges":[{"id":"e-de275f2bc42bbd59","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-4182d3b7552bf009","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":"d4a244c02036db20ddb82733a8a68957fe306ee8f0c90f8afc72965c08cf9bef","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6aadffa116a44288","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","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:         9,         18;\n  2:        81,        288,         243;\n  3:       729,       3690,        5913,         2916;\n  4:      6561,      43056,      101250,        99144,        32805;\n  5:     59049,     476082,     1475982,      2169504,      1476225,       354294;\n  6:    531441,    5081904,    19561581,     38415384,     39963051,     20312856,      3720087;\n  7:   4782969,   52911594,   243314847,    598471092,    841113639,    663828858,    264126177,     38263752;\n  8:  43046721,  540774432,  2892033396,   8550680112,  15165029790,  16325867520,  10222799076,   3290682672,   387420489;\n  9: 387420489, 5448100626, 33219793548, 114752321568, 245966575590, 336358456740, 289739507436, 148539885264, 39646030041, 3874204890;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9+54*x+81*x^2)*P(n-2,x) + (10+18*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 18*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 9 + 18*x;\nP[n_, x_] := P[n, x] = Expand[-(9 + 54*x + 81*x^2)*P[n - 2, x] + (10 + 18*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; v[2] = 9 + 18*x; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 54*x + 81*x^2)*v[i - 2] + (10 + 18*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-c85abc56078671cc","steps":["basis A013610","basis A013610"],"edges":["e-12f73d6f0ea24cf5","e-de275f2bc42bbd59"],"text":"A232535 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A232535 \u2192 basis A013610 \u2192 basis A013610","chain":null}