{"id":"n-264b8b71d4b9f777","revision":0,"first_index":0,"fingerprint":"f6-fd12dfaec634c7e4","name":"basis_map_A008277 of basis_map_A013610 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["9","6"],["25","18"],["241","432","108"],["1041","2040","540"],["10681","39510","24300","3240"],["60649","248346","162540","22680"],["658785","4568928","5223960","1632960","136080"],["4540321","35294832","43150968","14152320","1224720"],["51972841","646554150","1230185880","678358800","128595600","7348320"],["415198521","5860249890","11978069400","6947020080","1369645200","80831520"],["4988808529","108859796496","325291536900","282149904960","91406977200","11639738880","484989120"],["44847866545","1123276071528","3621966184980","3312041166720","1117536179760","147113366400","6304858560"],["563683953561","21324886099062","96688829905668","124051448921400","61150531281840","13026888594720","1191618267840","37829151360"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","15","43","781","3621","77731","494215","12220713","98363161","2743015591","26651014611","824821752085","9373085694493","318035716175451"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-8","-6"],["8","6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-6"],["0","6"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196999,"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-04d94984ffaa0d0b"]},{"id":197000,"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-04d94984ffaa0d0b"]},{"id":197001,"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-04d94984ffaa0d0b"]}],"region_exact":{},"incoming_edges":[{"id":"e-04d94984ffaa0d0b","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-264b8b71d4b9f777","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"69f551fd1fcc2674e848a8328aad69e176a17fb6f3170af0baf1c5b88fca855e","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-b2d4a92113cb2b7b","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","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":[],"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:       1;\n  2:       9,        6;\n  3:      25,       18;\n  4:     241,      432,      108;\n  5:    1041,     2040,      540;\n  6:   10681,    39510,    24300,     3240;\n  7:   60649,   248346,   162540,    22680;\n  8:  658785,  4568928,  5223960,  1632960,  136080;\n  9: 4540321, 35294832, 43150968, 14152320, 1224720;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8*n-8+(6*n-6)*x)*P(n-2,x) + (6*n-6)*x*d/dx P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(8*n - 8 + (6*n - 6)*x)*P[n - 2, x] + (6*n - 6)*x*D[P[n - 2, x], 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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (8*n - 8 + (6*n - 6)*x)*v[i - 2] + (6*n - 6)*x*deriv(v[i - 2], 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-c914b69838ac70ec","steps":["basis A013610","basis A008277"],"edges":["e-3c0217934a464502","e-04d94984ffaa0d0b"],"text":"A059344 \u2192 basis A013610 \u2192 basis A008277"},"headline":"A059344 \u2192 basis A013610 \u2192 basis A008277","chain":null}
