{"id":"n-427c1bc69ab53dd8","revision":1,"first_index":0,"fingerprint":"f6-e1bf771772d68e00","name":"basis_map_A008277 of Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1"],["7","6"],["25","24"],["61","60"],["481","1200","360"],["2731","7770","2520"],["10417","30576","10080"],["91225","514584","393120","60480"],["681121","4461120","3704400","604800"],["3493711","23784750","20124720","3326400"],["33597961","393514440","579126240","212889600","19958400"],["303321877","4196445396","6746556960","2637835200","259459200"],["1938378625","28094028144","46132926840","18283224960","1816214400"]],"degrees":[0,0,0,1,1,1,2,2,2,3,3,3,4,4,4],"row_sums":["1","1","1","13","49","121","2041","13021","51073","1059409","9451441","50729581","1239086641","14143618633","96264772969"],"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"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","6"],["-9","-9"],["3","3"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","6"],["0","-9"],["0","3"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162366,"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-1818aebb1bb797ef"]},{"id":162367,"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-1818aebb1bb797ef"]},{"id":162368,"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-1818aebb1bb797ef"]}],"region_exact":{},"incoming_edges":[{"id":"e-1818aebb1bb797ef","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-427c1bc69ab53dd8","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":"3c3bba55b24ffebb1c6ca74006240809d3de93fe35aeb5a7c5e203418e38585b","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-b12c32f9a9e37f0c","role":"sequence","name":"Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.","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:     1;\n  3:     7,      6;\n  4:    25,     24;\n  5:    61,     60;\n  6:   481,   1200,    360;\n  7:  2731,   7770,   2520;\n  8: 10417,  30576,  10080;\n  9: 91225, 514584, 393120, 60480;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n^2-9*n+6+(3*n^2-9*n+6)*x)*P(n-3,x) + (3*n^2-9*n+6)*x*d/dx P(n-3,x) + P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n^2 - 9*n + 6 + (3*n^2 - 9*n + 6)*x)*P[n - 3, x] + (3*n^2 - 9*n + 6)*x*D[P[n - 3, 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; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (3*n^2 - 9*n + 6 + (3*n^2 - 9*n + 6)*x)*v[i - 3] + (3*n^2 - 9*n + 6)*x*deriv(v[i - 3], 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-b12c32f9a9e37f0c","steps":["basis A008277"],"edges":["e-1818aebb1bb797ef"],"text":"A118394 \u2192 basis A008277"},"headline":"A118394 \u2192 basis A008277","chain":null}