{"id":"n-75656e7178d4d4e4","revision":0,"first_index":0,"fingerprint":"f6-a039d5598c7c2c03","name":"reversal of basis_map_A013611 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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"],["4","3"],["32","44","15"],["384","752","488","105"],["6144","15488","14576","6072","945"],["122880","377344","461888","281776","85692","10395"],["2949120","10653696","15990784","12767168","5719696","1363476","135135"],["82575360","342540288","607547392","597342464","351659008","123972768","24235920","2027025"],["2642411520","12365070336","25264701440","29443264512","21407910144","9945331712","2883086496","476875440","34459425"],["95126814720","495348350976","1144465588224","1539986849792","1330106790912","764782234368","292752416384","71946159264","10301172660","654729075"],["3805072588800","21811597148160","56180938899456","85633251344384","85543995482112","58523531983872","27770523577088","9025647114624","1922916250944","242513788860","13749310575"],["167423193907200","1047226944061440","2973628045983744","5060024653840384","5733500582133760","4542547303378944","2567944273020928","1035850515316480","292198198677888","54897680481552","6182786809080","316234143225"],["8036313307545600","54452473162629120","168914819808755712","317221884533932032","401708644288430080","361379785115533312","236825007689478144","113919431560714240","39921776419450624","9940043630061696","1669215778874640","169748909101800","7905853580625"],["417888291992371200","3048509063760445440","10253787405446283264","21056238130600869888","29453840385414529024","29637882221795344384","22072154597972262912","12318085648773050368","5151757025950716928","1594770232088375040","355180398513147072","53895769302908880","4994324931941100","213458046676875"],["23401744351572787200","182835268038363709440","662618857554044780544","1476509170071590928384","2260045967370638852096","2513882775597560627200","2095539241918511710208","1329905279672486445056","645722877267658608640","238708086749519793152","66138439047299112192","13318394637844162368","1842659505973059120","156789073640196900","6190283353629375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","91","1729","43225","1339975","49579075","2131900225","104463111025","5745471106375","350473737488875","23481740411754625","1714167050058087625","135419196954588922375","11510631741140058401875"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191518,"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-9701074122dc91be"]},{"id":191519,"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-9701074122dc91be"]},{"id":191520,"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-9701074122dc91be"]}],"region_exact":{},"incoming_edges":[{"id":"e-9701074122dc91be","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-75656e7178d4d4e4","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":"4494b6a28780b9f99339c34b7192cb21e1b8ec31f80348e9f4a5bbc18772bae5","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-be7df2e3fd154f3c","role":"sequence","name":"basis_map_A013611 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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-7a3a2fb1d513e881","kind":"fingerprint_related","node_a_id":"n-be7df2e3fd154f3c","node_b_id":"n-75656e7178d4d4e4","fingerprint":"f6-a039d5598c7c2c03","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a039d5598c7c2c03"},"other_id":"n-be7df2e3fd154f3c","other":{"id":"n-be7df2e3fd154f3c","name":"basis_map_A013611 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","first_index":0,"fingerprint":"f6-a039d5598c7c2c03","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","4"],["15","44","32"]],"card_rows":[["1"],["3","4"],["15","44","32"],["105","488","752","384"]],"row_sums_preview":["1","7","91","1729","43225","1339975","49579075","2131900225"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7fd16a32c438d522","steps":["basis A013611"],"edges":["e-5d9e378754634336"],"text":"A059364 \u2192 basis A013611"},"headline":"A059364 \u2192 basis A013611","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:           1;\n  1:           4,            3;\n  2:          32,           44,            15;\n  3:         384,          752,           488,           105;\n  4:        6144,        15488,         14576,          6072,           945;\n  5:      122880,       377344,        461888,        281776,         85692,        10395;\n  6:     2949120,     10653696,      15990784,      12767168,       5719696,      1363476,       135135;\n  7:    82575360,    342540288,     607547392,     597342464,     351659008,    123972768,     24235920,     2027025;\n  8:  2642411520,  12365070336,   25264701440,   29443264512,   21407910144,   9945331712,   2883086496,   476875440,    34459425;\n  9: 95126814720, 495348350976, 1144465588224, 1539986849792, 1330106790912, 764782234368, 292752416384, 71946159264, 10301172660, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+(2*n+1)*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*n + 1)*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*n + 1)*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-7fd16a32c438d522","steps":["basis A013611","reverse"],"edges":["e-5d9e378754634336","e-9701074122dc91be"],"text":"A059364 \u2192 basis A013611 \u2192 reverse"},"headline":"A059364 \u2192 basis A013611 \u2192 reverse","chain":null}
