{"id":"n-596387217b57e703","revision":0,"first_index":0,"fingerprint":"f6-ba46f4556b8fa9cf","name":"basis_map_A013609 of basis_map_A013610 of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["7","6"],["13","12"],["55","90","36"],["133","240","108"],["463","1110","864","216"],["1261","3348","2952","864"],["4039","12786","14796","7344","1296"],["11605","40440","52596","30240","6480"],["35839","141390","218088","163080","58320","7776"],["105469","453660","776304","660096","278640","46656"],["320503","1517034","2933172","2947104","1607040","443232","46656"],["953317","4871808","10312956","11565504","7239456","2395008","326592"],["2876335","15897030","37014192","46847160","34564320","14696640","3265920","279936"],["8596237","50847780","128122776","178117920","147394080","72503424","19595520","2239488"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["1","13","25","181","481","2653","8425","40261","141361","624493","2320825","9814741","37664641","155441533","607417225"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["7","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199209,"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-a02f54dc18f1c145"]},{"id":199210,"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-a02f54dc18f1c145"]},{"id":199211,"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-a02f54dc18f1c145"]},{"id":199212,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-a02f54dc18f1c145"]}],"region_exact":{},"incoming_edges":[{"id":"e-a02f54dc18f1c145","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-596387217b57e703","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"37f39080b10cb118ee221702d208c78c447f7b7d10105fb5d51fb1e01e7f6023","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9020569c3245b2bd","role":"sequence","name":"basis_map_A013610 of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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":[{"id":"l-801d0ad298309f3a","kind":"fingerprint_related","node_a_id":"n-3f9fff45963528a2","node_b_id":"n-596387217b57e703","fingerprint":"f6-ba46f4556b8fa9cf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ba46f4556b8fa9cf"},"other_id":"n-3f9fff45963528a2","other":{"id":"n-3f9fff45963528a2","name":"shift_x_plus_one of gamma_polynomial of Pascal-(1,6,1) array.","first_index":0,"fingerprint":"f6-ba46f4556b8fa9cf","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["7","6"]],"card_rows":[["1"],["1"],["7","6"],["13","12"]],"row_sums_preview":["1","1","13","25","181","481","2653","8425"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e189038e95d51606","steps":["basis A007318","basis A013609"],"edges":["e-3091186ded9433db","e-30a39ecb8a3fcb98"],"text":"A304236 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A304236 \u2192 basis A007318 \u2192 basis A013609","chain":null}}],"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:     7,      6;\n  2:    13,     12;\n  3:    55,     90,     36;\n  4:   133,    240,    108;\n  5:   463,   1110,    864,    216;\n  6:  1261,   3348,   2952,    864;\n  7:  4039,  12786,  14796,   7344,  1296;\n  8: 11605,  40440,  52596,  30240,  6480;\n  9: 35839, 141390, 218088, 163080, 58320, 7776;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+6*x)*P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 7 + 6*x;\nP[n_, x_] := P[n, x] = Expand[(6 + 6*x)*P[n - 2, 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] = 7 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (6 + 6*x)*v[i - 2] + 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-ac04e52510a48bb5","steps":["basis A013610","basis A013609"],"edges":["e-6895a0e432d3b559","e-a02f54dc18f1c145"],"text":"A249139 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A249139 \u2192 basis A013610 \u2192 basis A013609","chain":null}