{"id":"n-fdb150e76bed506a","revision":0,"first_index":0,"fingerprint":"f6-7d794259d02ffafb","name":"reversal of Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 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":false,"rows":[["1"],["4","1"],["16","24","1"],["64","240","124","1"],["256","1792","2656","624","1"],["1024","11520","33920","26400","3124","1"],["4096","67584","331520","546560","250096","15624","1"],["16384","372736","2745344","7822080","8105664","2313360","78124","1"],["65536","1966080","20299776","88940544","165398016","114627072","21132736","390624","1"],["262144","10027008","138215424","863256576","2529343488","3270274560","1574682880","191757120","1953124","1"],["1048576","49807360","883752960","7461273600","31698788352","66197311488","61893399040","21237905920","1733626576","9765624","1"],["4194304","242221056","5377884160","59008942080","343172087808","1057258954752","1637717137408","1137139407360","283027283264","15641701680","48828124","1"],["16777216","1157627904","31442599936","435017482240","3319983439872","14181026365440","32982342418432","38940617515008","20463479058176","3741921489152","140970627616","244140624","1"],["67108864","5452595200","177863655424","3029216526336","29375580602368","166283558977536","543179134271488","980321030520832","899606884047872","362846829945600","49208861869440","1269712211040","1220703124","1"],["268435456","25367150592","978631786496","20120730599424","241700199989248","1752030718197760","7660073983344640","19673479015956480","28106453299212288","20343131884787712","6365231556552960","644794053146880","11432292711856","6103515624","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","41","429","5329","75989","1215481","21453693","412820385","8579772325","191166679497","4538638641997","114238219541617","3035305413035125","84819458105387417"],"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":[["4","-3"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187074,"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-5d46abdb65d31413"]},{"id":187075,"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-5d46abdb65d31413"]},{"id":187076,"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-5d46abdb65d31413"]}],"region_exact":{},"incoming_edges":[{"id":"e-5d46abdb65d31413","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-fdb150e76bed506a","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":"53d51bc0befe31ad0ee216169c104c4f08dee1f4d4bec39c67a838fdf0e795f6","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-4c2a0f8cb422ddaa","role":"sequence","name":"Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 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-46869d7e66eccadc","kind":"fingerprint_related","node_a_id":"n-4c2a0f8cb422ddaa","node_b_id":"n-fdb150e76bed506a","fingerprint":"f6-7d794259d02ffafb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7d794259d02ffafb"},"other_id":"n-4c2a0f8cb422ddaa","other":{"id":"n-4c2a0f8cb422ddaa","name":"Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).","first_index":0,"fingerprint":"f6-7d794259d02ffafb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","24","16"]],"card_rows":[["1"],["1","4"],["1","24","16"],["1","124","240","64"]],"row_sums_preview":["1","5","41","429","5329","75989","1215481","21453693"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A285061"],"derivation":{"root":"n-4c2a0f8cb422ddaa","steps":[],"edges":[],"text":"A285061"},"headline":"A285061: Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).","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,        1;\n  2:     16,       24,         1;\n  3:     64,      240,       124,         1;\n  4:    256,     1792,      2656,       624,          1;\n  5:   1024,    11520,     33920,     26400,       3124,          1;\n  6:   4096,    67584,    331520,    546560,     250096,      15624,          1;\n  7:  16384,   372736,   2745344,   7822080,    8105664,    2313360,      78124,         1;\n  8:  65536,  1966080,  20299776,  88940544,  165398016,  114627072,   21132736,    390624,       1;\n  9: 262144, 10027008, 138215424, 863256576, 2529343488, 3270274560, 1574682880, 191757120, 1953124, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+(4*n-3)*x)*P(n-1,x) - 4*x^2*d/dx 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 + (4*n - 3)*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], 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 + (4*n - 3)*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], x)); 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-4c2a0f8cb422ddaa","steps":["reverse"],"edges":["e-5d46abdb65d31413"],"text":"A285061 \u2192 reverse"},"headline":"A285061 \u2192 reverse","chain":null}
