{"id":"n-11a69976f40ceb7e","revision":1,"first_index":0,"fingerprint":"f6-fd3b298a2ad2f471","name":"Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), 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"],["27"],["81","1"],["243","6"],["729","27"],["2187","108"],["6561","405","1"],["19683","1458","9"],["59049","5103","54"],["177147","17496","270"],["531441","59049","1215","1"],["1594323","196830","5103","12"],["4782969","649539","20412","90"]],"degrees":[0,0,0,0,1,1,1,1,2,2,2,2,3,3,3],"row_sums":["1","3","9","27","82","249","756","2295","6967","21150","64206","194913","591706","1796268","5453010"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A318773","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3"],["9"],["27"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160440,"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-e3a638b293d2df1c"]},{"id":160441,"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-e3a638b293d2df1c"]},{"id":160442,"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-e3a638b293d2df1c"]}],"region_exact":{},"incoming_edges":[{"id":"e-e3a638b293d2df1c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1/243\",\"intercept\":\"0\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-11a69976f40ceb7e","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"c1dd4a40d0bfbf4a0691ad50d0ea8fce3dd426bf04ef3c1061ed0ab7eecb28a7","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":[{"id":"e-063b555ef8572acb","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-4126579c1198cdeb","conclusion_name":"basis_map_A013610 of Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-49214357343b38a8","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-544767382379f5dd","conclusion_name":"basis_map_A008277 of Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-8abfc1e62fa10a1f","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-720f52095a098452","conclusion_name":"basis_map_A013611 of Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-b1356930ce9c898c","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-84604e10f166b874","conclusion_name":"basis_map_A013609 of Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-fc673ff7432a109d","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-905d1fc5d3865cdd","conclusion_name":"basis_map_A007318 of Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-dbbb5664ea9ec441","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-905d1fc5d3865cdd","conclusion_name":"basis_map_A007318 of Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows."}],"used_by_total":6,"links":[{"id":"l-232ecb512df3cb59","kind":"oeis_match","node_a_id":"n-11a69976f40ceb7e","node_b_id":null,"fingerprint":null,"oeis_id":"A318773","in_catalog":1,"target":"triangle","row_alignment":"node_lengths","rows_matched":15,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":15,"target":"triangle","terms_compared":36},"in_catalog":true,"oeis_id":"A318773","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-8e269eea4e113f5b","kind":"oeis_match","node_a_id":"n-11a69976f40ceb7e","node_b_id":null,"fingerprint":null,"oeis_id":"A052917","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A052917","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A318773","relation":"identification"},{"oeis_id":"A052917","relation":"match_row_sums"},{"oeis_id":"A318773","relation":"match_triangle"}],"bfiles":[{"node_id":"n-11a69976f40ceb7e","oeis_id":"A052917","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":26,"oeis_terms_available":26,"term_count":200,"raw_size":10584},{"node_id":"n-11a69976f40ceb7e","oeis_id":"A318773","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":87,"terms_matched":36,"oeis_terms_agreeing":50,"oeis_terms_available":50,"term_count":990,"raw_size":23892}],"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:    27;\n  4:    81,    1;\n  5:   243,    6;\n  6:   729,   27;\n  7:  2187,  108;\n  8:  6561,  405, 1;\n  9: 19683, 1458, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 3*P(n-1,x) + x*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3, P(2,x) = 9, P(3,x) = 27.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[2, x_] := 9;\nP[3, x_] := 27;\nP[n_, x_] := P[n, x] = Expand[3*P[n - 1, x] + x*P[n - 4, 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; v[4] = 27; for(n = 4, N, my(i = n + 1); v[i] = 3*v[i - 1] + x*v[i - 4]); 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}},{"oeis_id":"A318773","target":"triangle","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:    27;\n  4:    81,    1;\n  5:   243,    6;\n  6:   729,   27;\n  7:  2187,  108;\n  8:  6561,  405, 1;\n  9: 19683, 1458, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 3*P(n-1,x) + x*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3, P(2,x) = 9, P(3,x) = 27.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[2, x_] := 9;\nP[3, x_] := 27;\nP[n_, x_] := P[n, x] = Expand[3*P[n - 1, x] + x*P[n - 4, 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; v[4] = 27; for(n = 4, N, my(i = n + 1); v[i] = 3*v[i - 1] + x*v[i - 4]); 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":["x \u21a6 (1/243)x"],"edges":["e-e3a638b293d2df1c"],"text":"A318772 \u2192 x \u21a6 (1/243)x"},"headline":"A318773: Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.","chain":"A318772 \u2192 x \u21a6 (1/243)x"}