{"id":"n-77311ba2b23da2eb","revision":1,"first_index":0,"fingerprint":"f6-fe2eac7fb802bf9f","name":"basis_map_A013610 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","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":true,"rows":[["1"],["1","3"],["9","36","27"],["225","1035","1215","405"],["11025","55440","81270","45360","8505"],["893025","4788315","8079750","5868450","1913625","229635"],["108056025","608855940","1135664145","976714200","425207475","90935460","7577955"],["18261468225","107110838835","215672622165","209355601455","109951917075","31951184265","4827157335","295540245"],["4108830350625","24921704808000","53346327734700","56810278324800","34160183407350","12136852728000","2523913692300","283718635200","13299311025"],["1187451971330625","7411923037393875","16688095660536300","19138833150336900","12769617199288950","5249719792166850","1348390546202700","210714283880100","18313151281425","678264862275"],["428670161650355625","2743388978865034500","6446882146585055175","7860340219922190000","5700745298512514250","2623017025331703000","786003015332685150","152926117614270000","18621761793760125","1288703238322500","38661097149675"],["189043541287806830625","1236840759863452618875","3015908532312506505675","3872563612220544266025","3009230110499116754250","1515897461977569420750","511877402357611440150","116958607833852234450","17846542360747225125","1741489121107110375","98237847857324175","2435649120429525"],["100004033341249813400625","667332766316625106698000","1680757626023894172204450","2256683839594230865618800","1859089617697250317353975","1009546635010573279620000","375380070723628741871100","97190644306783021394400","17510964849371086268175","2152659167957219922000","172130570872915104450","8066869886862586800","168059789309637225"],["62502520838281133375390625","424583281448484427691296875","1100523473738680740630131250","1536484221698186353927083750","1331182299030348763267644375","770398368208902073564048125","310328541828060959640937500","88897657996011544011832500","18233651353865655522189375","2658734343676093921363125","269031044392363434431250","17951586494757749583750","710052609833217275625","12604484198222791875"],["45564337691106946230659765625","314583916363845919590362062500","836672858152825498562360728125","1209239398990810992003884685000","1094887117950677343090206533125","669446176645747861452870277500","288631774817577507536931335625","89943004567165353315541830000","20493042134644997940634486875","3415143096202990565971057500","411481113199796551330794375","34878221150459837635485000","1971706858643793110214375","66702930376995014602500","1020963220056046141875"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","72","2880","201600","21772800","3353011200","697426329600","188305108992000","64023737057280000","26761922089943040000","13488008733331292160000","8065829222532112711680000","5646080455772478898176000000","4573325169175707907522560000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-3"],["-4","6"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167585,"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-afc618911e5721d2"]},{"id":167586,"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-afc618911e5721d2"]},{"id":167587,"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-afc618911e5721d2"]}],"region_exact":{},"incoming_edges":[{"id":"e-afc618911e5721d2","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-77311ba2b23da2eb","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":"df354c2ac126062923a0be95874fe1da678a644be8265dd3b452b8ac24d21151","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-1336d058dca9979e","role":"sequence","name":"Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","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-fccc28dd0b726f87","kind":"oeis_match","node_a_id":"n-77311ba2b23da2eb","node_b_id":null,"fingerprint":null,"oeis_id":"A327882","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A327882","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A327882","relation":"match_row_sums"}],"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:                1,                3;\n  2:                9,               36,                27;\n  3:              225,             1035,              1215,               405;\n  4:            11025,            55440,             81270,             45360,              8505;\n  5:           893025,          4788315,           8079750,           5868450,           1913625,           229635;\n  6:        108056025,        608855940,        1135664145,         976714200,         425207475,         90935460,          7577955;\n  7:      18261468225,     107110838835,      215672622165,      209355601455,      109951917075,      31951184265,       4827157335,       295540245;\n  8:    4108830350625,   24921704808000,    53346327734700,    56810278324800,    34160183407350,   12136852728000,    2523913692300,    283718635200,    13299311025;\n  9: 1187451971330625, 7411923037393875, 16688095660536300, 19138833150336900, 12769617199288950, 5249719792166850, 1348390546202700, 210714283880100, 18313151281425, 678264862275;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n^2-4*n+1+(6*n-3)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4*n^2 - 4*n + 1 + (6*n - 3)*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; for(n = 1, N, my(i = n + 1); v[i] = (4*n^2 - 4*n + 1 + (6*n - 3)*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-1336d058dca9979e","steps":["basis A013610"],"edges":["e-afc618911e5721d2"],"text":"A204420 \u2192 basis A013610"},"headline":"A204420 \u2192 basis A013610","chain":null}
