{"id":"n-ea8536338ecfd7fc","revision":0,"first_index":0,"fingerprint":"f6-c748c34fca57e6cb","name":"reversal of basis_map_A008277 of Riordan array (1-x,x/(1-x)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1","3","1"],["1","7","10","2"],["1","12","38","32","4"],["1","18","98","184","100","8"],["1","25","206","674","836","308","16"],["1","33","381","1910","4206","3652","940","32"],["1","42","645","4577","15666","24682","15548","2852","64"],["1","52","1023","9737","47705","118678","138958","65044","8620","128"],["1","63","1543","18944","125601","452613","851026","759834","268796","25988","256"],["1","75","2236","34374","296097","1457421","4019317","5865990","4067966","1101172","78220","512"],["1","88","3136","58970","639837","4122294","15678685","34001209","39263906","21441002","4482908","235172","1024"],["1","102","4280","96602","1288507","10520664","52779331","159430689","277272369","257024438","111687918","18166804","706540","2048"],["1","117","5708","152242","2447731","24694241","157985971","634444668","1552717881","2197931021","1653834546","576606394","73373756","2121668","4096"],["1","133","7463","232154","4426877","54067013","429622622","2214304378","7262719893","14619674069","17039351693","10499613670","2956405726","295616692","6369100","8192"],["1","150","9591","344099","7677033","111616414","1078426778","6940153220","29405763673","79984153106","133996744245","129775075521","65954087746","15077645322","1188835868","19115492","16384"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["5","20","87","409","2066","11155","64077","389946","2504665","16923381","119928232","888980293","6876320041","55382419676","463539664643"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180904,"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-46389ff40ebfa6f7"]},{"id":180905,"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-46389ff40ebfa6f7"]},{"id":180906,"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-46389ff40ebfa6f7"]}],"region_exact":{},"incoming_edges":[{"id":"e-46389ff40ebfa6f7","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-ea8536338ecfd7fc","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":"96e29be880d668ffc0a55e5e1dbb3a7b6919b499b92d5506e7c4259a7517734c","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-9a6b0e2632270c41","role":"sequence","name":"basis_map_A008277 of Riordan array (1-x,x/(1-x)).","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-a7b50b5e8a688d6b","kind":"fingerprint_related","node_a_id":"n-9a6b0e2632270c41","node_b_id":"n-ea8536338ecfd7fc","fingerprint":"f6-c748c34fca57e6cb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c748c34fca57e6cb"},"other_id":"n-9a6b0e2632270c41","other":{"id":"n-9a6b0e2632270c41","name":"basis_map_A008277 of Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-c748c34fca57e6cb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1","3","1"],["2","10","7","1"],["4","32","38","12","1"]],"card_rows":[["1","3","1"],["2","10","7","1"],["4","32","38","12","1"],["8","100","184","98","18","1"]],"row_sums_preview":["5","20","87","409","2066","11155","64077","389946"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":["basis A008277"],"edges":["e-94a89e959b0863ff"],"text":"A159854 \u2192 basis A008277"},"headline":"A159854 \u2192 basis A008277","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,  3,    1;\n  1: 1,  7,   10,     2;\n  2: 1, 12,   38,    32,      4;\n  3: 1, 18,   98,   184,    100,       8;\n  4: 1, 25,  206,   674,    836,     308,      16;\n  5: 1, 33,  381,  1910,   4206,    3652,     940,      32;\n  6: 1, 42,  645,  4577,  15666,   24682,   15548,    2852,      64;\n  7: 1, 52, 1023,  9737,  47705,  118678,  138958,   65044,    8620,     128;\n  8: 1, 63, 1543, 18944, 125601,  452613,  851026,  759834,  268796,   25988,   256;\n  9: 1, 75, 2236, 34374, 296097, 1457421, 4019317, 5865990, 4067966, 1101172, 78220, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n+3)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 3*x + x^2.","mathematica":"P[0, x_] := 1 + 3*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (n + 3)*x)*P[n - 1, x] - 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 + 3*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (n + 3)*x)*v[i - 1] - 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-7717563017cb9e5f","steps":["basis A008277","reverse"],"edges":["e-94a89e959b0863ff","e-46389ff40ebfa6f7"],"text":"A159854 \u2192 basis A008277 \u2192 reverse"},"headline":"A159854 \u2192 basis A008277 \u2192 reverse","chain":null}