{"id":"n-40fdee53f6f3666d","revision":0,"first_index":0,"fingerprint":"f6-f8d027db0af7f231","name":"reversal of basis_map_A013610 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"],["3","3"],["18","33","15"],["162","423","366","105"],["1944","6534","8199","4554","945"],["29160","119394","194859","158499","64269","10395"],["524880","2528172","5059584","5386149","3217329","1022607","135135"],["11022480","60964812","144173844","189002889","148356144","69734682","18176940","2027025"],["264539520","1650537648","4496574060","6987024684","6773596569","4195686816","1621736154","357656580","34459425"],["7142567040","49590767376","152767714932","274084573608","315640576359","241981878843","123504925662","40469714586","7725879495","654729075"],["214277011200","1637716929120","5624437562856","11430659221812","15224993336538","13887908468829","8786767225563","3807694876482","1081640391156","181885341645","13749310575"],["7071141369600","58973029918560","223273928944008","506573818265484","765329942207430","808475826211731","609385213226646","327749577111855","123271115067234","30879945270873","4637090106810","316234143225"],["254561089305600","2299807611308160","9512187189948288","23818505681157624","40216223376104580","48238378298808066","42149763331452531","27033615106692930","12631499570216799","4193455906432278","938933875616985","127311681826350","7905853580625"],["9927882482918400","96565646252269440","433070105913303552","1185750775693751112","2211532365059334468","2967134784808338234","2946276983994466491","2192354599110242607","1222536091119164271","504595268746712415","149841730622733921","30316370232886245","3745743698955825","213458046676875"],["416971064282572800","4343665734599950080","20989348189674562944","62360565650623349712","127271131827610829904","188754099548670905400","209790542087209401408","177520925698469717733","114924799201201934985","56646547929817685289","20926615479809484717","5618697737840505999","1036495972109845755","117591805230147675","6190283353629375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","66","1056","22176","576576","17873856","643458816","26381811456","1213563326976","61891729675776","3465936861843456","211422148572450816","13953861805781753856","990724188210504523776"],"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"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190514,"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-35ec3b218b9f2497"]},{"id":190515,"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-35ec3b218b9f2497"]},{"id":190516,"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-35ec3b218b9f2497"]}],"region_exact":{},"incoming_edges":[{"id":"e-35ec3b218b9f2497","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-40fdee53f6f3666d","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":"9437a1be605375e7fbb4113ad49896cfc681d160d4020b367d5150d9ace9fa4c","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-629fc911ba71fa1d","role":"sequence","name":"basis_map_A013610 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-84a31ad2b1f04488","kind":"fingerprint_related","node_a_id":"n-629fc911ba71fa1d","node_b_id":"n-40fdee53f6f3666d","fingerprint":"f6-f8d027db0af7f231","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f8d027db0af7f231"},"other_id":"n-629fc911ba71fa1d","other":{"id":"n-629fc911ba71fa1d","name":"basis_map_A013610 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-f8d027db0af7f231","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","3"],["15","33","18"]],"card_rows":[["1"],["3","3"],["15","33","18"],["105","366","423","162"]],"row_sums_preview":["1","6","66","1056","22176","576576","17873856","643458816"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7fd16a32c438d522","steps":["basis A013610"],"edges":["e-07b12f78e33af42b"],"text":"A059364 \u2192 basis A013610"},"headline":"A059364 \u2192 basis A013610","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:          3,           3;\n  2:         18,          33,           15;\n  3:        162,         423,          366,          105;\n  4:       1944,        6534,         8199,         4554,          945;\n  5:      29160,      119394,       194859,       158499,        64269,        10395;\n  6:     524880,     2528172,      5059584,      5386149,      3217329,      1022607,       135135;\n  7:   11022480,    60964812,    144173844,    189002889,    148356144,     69734682,     18176940,     2027025;\n  8:  264539520,  1650537648,   4496574060,   6987024684,   6773596569,   4195686816,   1621736154,   357656580,   34459425;\n  9: 7142567040, 49590767376, 152767714932, 274084573608, 315640576359, 241981878843, 123504925662, 40469714586, 7725879495, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*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[(3*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] = (3*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 A013610","reverse"],"edges":["e-07b12f78e33af42b","e-35ec3b218b9f2497"],"text":"A059364 \u2192 basis A013610 \u2192 reverse"},"headline":"A059364 \u2192 basis A013610 \u2192 reverse","chain":null}