{"id":"n-84ac0e69bd31191f","revision":1,"first_index":0,"fingerprint":"f6-a889c7483457dc54","name":"basis_map_A008277 of Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.","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"],["1"],["4","3"],["7","6"],["10","9"],["13","12"],["25","42","9"],["46","99","27"],["76","183","54"],["115","294","90"],["190","621","297","27"],["328","1353","837","108"],["556","2679","1872","270"]],"degrees":[0,0,0,0,1,1,1,1,2,2,2,2,3,3,3],"row_sums":["1","1","1","1","7","13","19","25","76","172","313","499","1135","2626","5377"],"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"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162129,"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-95ffdd039009de97"]},{"id":162130,"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-95ffdd039009de97"]},{"id":162131,"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-95ffdd039009de97"]}],"region_exact":{},"incoming_edges":[{"id":"e-95ffdd039009de97","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-84ac0e69bd31191f","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":"fc194b6b08027f46cfd52684c57436bfb3cd52b31283a71b366e9cb5d6b286f9","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-a2db6b6ded7a955c","role":"sequence","name":"Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.","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:  1;\n  4:  4,  3;\n  5:  7,  6;\n  6: 10,  9;\n  7: 13, 12;\n  8: 25, 42,  9;\n  9: 46, 99, 27;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+3*x)*P(n-4,x) + 3*x*d/dx P(n-4,x) + P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1, P(3,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 4, x] + 3*x*D[P[n - 4, 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; v[4] = 1; for(n = 4, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 4] + 3*x*deriv(v[i - 4], 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-a2db6b6ded7a955c","steps":["basis A008277"],"edges":["e-95ffdd039009de97"],"text":"A318772 \u2192 basis A008277"},"headline":"A318772 \u2192 basis A008277","chain":null}