{"id":"n-928345a971b12a92","revision":1,"first_index":0,"fingerprint":"f6-585b6db43d3cef9f","name":"basis_map_A008277 of Triangle T(n,k) = 3*T(n-1,k) +  T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or 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"],["3"],["9"],["28","1"],["87","6"],["270","27"],["838","111","1"],["2601","432","9"],["8073","1620","54"],["25057","5920","276","1"],["77772","21225","1287","12"],["241389","74988","5643","90"],["749224","261861","23677","550","1"],["2325444","905805","96117","2985","15"],["7217721","3108780","380268","14958","135"]],"degrees":[0,0,0,1,1,1,2,2,2,3,3,3,4,4,4],"row_sums":["1","3","9","29","93","297","950","3042","9747","31254","100296","322110","1035313","3330366","10721862"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3"],["9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162330,"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-f7cac2f9abe8b6c3"]},{"id":162331,"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-f7cac2f9abe8b6c3"]},{"id":162332,"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-f7cac2f9abe8b6c3"]}],"region_exact":{},"incoming_edges":[{"id":"e-f7cac2f9abe8b6c3","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-928345a971b12a92","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":"41c2bbc182a69de0a2c37052a5fe960c369012eb41a14c980c4e51446c967e39","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-a5b33928f20c14a2","role":"sequence","name":"Triangle T(n,k) = 3*T(n-1,k) +  T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or 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:     3;\n  2:     9;\n  3:    28,    1;\n  4:    87,    6;\n  5:   270,   27;\n  6:   838,  111,   1;\n  7:  2601,  432,   9;\n  8:  8073, 1620,  54;\n  9: 25057, 5920, 276, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-3,x) + x*d/dx P(n-3,x) + 3*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3, P(2,x) = 9.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[2, x_] := 9;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 3, x] + x*D[P[n - 3, x], x] + 3*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] = 3; v[3] = 9; for(n = 3, N, my(i = n + 1); v[i] = (1 + x)*v[i - 3] + x*deriv(v[i - 3], x) + 3*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-0d55e0bb3c384646","steps":["x \u21a6 (1/27)x","basis A008277"],"edges":["e-b8d37b5fe37403cf","e-f7cac2f9abe8b6c3"],"text":"A102547 \u2192 x \u21a6 (1/27)x \u2192 basis A008277"},"headline":"A102547 \u2192 x \u21a6 (1/27)x \u2192 basis A008277","chain":null}