{"id":"n-bb62cc8228f1a151","revision":0,"first_index":0,"fingerprint":"f6-c54a9b6c897ea5d0","name":"reversal of basis_map_A013611 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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","2"],["32","28","6"],["384","464","184","24"],["6144","9344","5264","1304","120"],["122880","223744","161344","57664","10224","720"],["2949120","6230016","5438464","2513344","649024","88848","5040"],["82575360","198033408","202117120","113881344","38279424","7679936","851904","40320"],["2642411520","7080247296","8250048512","5463257088","2249873664","590272768","96380352","8957376","362880"],["95126814720","281313017856","367804219392","279177740288","135628022784","43748556288","9372420352","1286269056","102637440","3628800"],["3805072588800","12298915676160","17806611972096","15212956024832","8496076054528","3241850502144","856130933248","154547386112","18254457216","1274163840","39916800"],["167423193907200","586813160816640","931077914886144","883049408757760","556382818697216","244594334748672","76571967088640","17073656187904","2657764750848","275116695552","17046305280","479001600"],["8036313307545600","30343533239992320","52320311005151232","54490384513892352","38186017611317248","18973504711000064","6855180771987456","1814971069171712","349530238483456","47756543147520","4394739695616","244594045440","6227020800"],["417888291992371200","1690372114785239040","3145465637627756544","3565984348794519552","2748538298982989824","1521226491530444800","622098466097348608","190351026404753408","43585167369543680","7376763582439424","897118068237312","74245246101504","3748121717760","87178291200"],["23401744351572787200","100929162807858954240","201501657428932952064","246877108096909443072","207407909974965223424","126416758010449756160","57655911474408194048","19991134470126419968","5296034768765747200","1066876271159762944","160890065557880832","17614504805243904","1323573507717120","61103810073600","1307674368000"]],"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"],["4","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190774,"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-c2ff08a207cf246f"]},{"id":190775,"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-c2ff08a207cf246f"]},{"id":190776,"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-c2ff08a207cf246f"]}],"region_exact":{},"incoming_edges":[{"id":"e-c2ff08a207cf246f","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-bb62cc8228f1a151","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":"d495a8ed49aa4ff32e32b9054af9612ed5120304867263a4843cb2006fd6860e","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-e63fe40b997d2bc1","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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-8e5bfa6dd93c06ef","kind":"fingerprint_related","node_a_id":"n-bb62cc8228f1a151","node_b_id":"n-a2aa153b0ee1e677","fingerprint":"f6-c54a9b6c897ea5d0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c54a9b6c897ea5d0"},"other_id":"n-a2aa153b0ee1e677","other":{"id":"n-a2aa153b0ee1e677","name":"basis_map_A013611 of reversal of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-c54a9b6c897ea5d0","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["2","4"]],"card_rows":[["1"],["1"],["2","4"],["6","28","32"]],"row_sums_preview":["1","1","6","66","1056","22176","576576","17873856"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["reverse","basis A013611"],"edges":["e-c7203dc00a9aba0b","e-7a39a79de7b33247"],"text":"A132393 \u2192 reverse \u2192 basis A013611"},"headline":"A132393 \u2192 reverse \u2192 basis A013611","chain":null}},{"id":"l-c022e8b0d9801f9f","kind":"fingerprint_related","node_a_id":"n-e63fe40b997d2bc1","node_b_id":"n-bb62cc8228f1a151","fingerprint":"f6-c54a9b6c897ea5d0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c54a9b6c897ea5d0"},"other_id":"n-e63fe40b997d2bc1","other":{"id":"n-e63fe40b997d2bc1","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","first_index":0,"fingerprint":"f6-c54a9b6c897ea5d0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["6","28","32"]],"card_rows":[["1"],["2","4"],["6","28","32"],["24","184","464","384"]],"row_sums_preview":["1","6","66","1056","22176","576576","17873856","643458816"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4d60c3dd8627c01d","steps":["basis A013611"],"edges":["e-60ae4909e1eb1035"],"text":"A094638 \u2192 basis A013611"},"headline":"A094638 \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,            2;\n  2:          32,           28,            6;\n  3:         384,          464,          184,           24;\n  4:        6144,         9344,         5264,         1304,          120;\n  5:      122880,       223744,       161344,        57664,        10224,         720;\n  6:     2949120,      6230016,      5438464,      2513344,       649024,       88848,       5040;\n  7:    82575360,    198033408,    202117120,    113881344,     38279424,     7679936,     851904,      40320;\n  8:  2642411520,   7080247296,   8250048512,   5463257088,   2249873664,   590272768,   96380352,    8957376,    362880;\n  9: 95126814720, 281313017856, 367804219392, 279177740288, 135628022784, 43748556288, 9372420352, 1286269056, 102637440, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+(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 + (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 + (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-4d60c3dd8627c01d","steps":["basis A013611","reverse"],"edges":["e-60ae4909e1eb1035","e-c2ff08a207cf246f"],"text":"A094638 \u2192 basis A013611 \u2192 reverse"},"headline":"A094638 \u2192 basis A013611 \u2192 reverse","chain":null}