{"id":"n-1a0f876fa1441737","revision":1,"first_index":0,"fingerprint":"f6-bc84b3931f09589a","name":"basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 1","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["0","1"],["0","3","1"],["0","18","9","1"],["0","162","99","18","1"],["0","1944","1350","315","30","1"],["0","29160","22194","6075","765","45","1"],["0","524880","428652","131544","19845","1575","63","1"],["0","11022480","9526572","3191076","548289","52920","2898","84","1"],["0","264539520","239660208","86112396","16350012","1818369","122472","4914","108","1"],["0","7142567040","6735365136","2564694900","527562720","65445975","5125113","255150","7830","135","1"],["0","214277011200","209203521120","83676212136","18391576500","2490941970","219199365","12779613","490050","11880","165","1"],["0","7071141369600","7117993208160","2970518521608","690598236636","100592661510","9724521015","640926594","28951263","882090","17325","198","1"],["0","254561089305600","263318896863360","114056659986048","27832055040504","4311934050996","450675418050","32797878399","1683172062","60706503","1505790","24453","234","1"],["0","9927882482918400","10523998066976640","4711528636319232","1199506806565704","195997483029348","21888275354946","1729792675611","98441588817","4050725679","119432313","2459457","33579","273","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","4","28","280","3640","58240","1106560","24344320","608608000","17041024000","528271744000","17961239296000","664565853952000","26582634158080000"],"nonn":true,"definitions":[{"position":0,"source":"derived:root_bound_shift:derived:basis_map_coefficients_of:oeis_catalog:A371080","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","1"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160465,"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-aa6a1f197b91b3b6"]},{"id":160466,"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-aa6a1f197b91b3b6"]},{"id":160467,"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-aa6a1f197b91b3b6"]}],"region_exact":{},"incoming_edges":[{"id":"e-aa6a1f197b91b3b6","operation":{"operation":"parent_provers","route":"root_domain"},"operation_detail":"root_domain","conclusion_id":"n-1a0f876fa1441737","premise_count":0,"theorem":"polytool parent provers, route root_domain: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"b7c12a3b9ae25a3e4d7981f9e3d38ff781f8d0f558a8fde84bfbc63dfe4e014d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[{"id":"e-1a9b6c8547b01dec","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-6dd41086947f478d","conclusion_name":"basis_map_A008277 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 1"},{"id":"e-8877bf17db51e7c6","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-b0592fccca597966","conclusion_name":"basis_map_A013610 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 1"},{"id":"e-c271a57d901f1e05","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-1a01d480c8d600c3","conclusion_name":"basis_map_A013611 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 1"},{"id":"e-d3f9f2e05c32db01","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-20bccc4c38847de1","conclusion_name":"basis_map_A013609 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 1"},{"id":"e-fe032cf3f375eb62","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-260e022530e5976c","conclusion_name":"Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle."},{"id":"e-c5f7781ad297615f","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-2e038ac8288fcb6a","conclusion_name":"basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3."}],"used_by_total":6,"links":[{"id":"l-457faa80ae722431","kind":"fingerprint_related","node_a_id":"n-1a0f876fa1441737","node_b_id":"n-b6cbf920aaa38713","fingerprint":"f6-bc84b3931f09589a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bc84b3931f09589a"},"other_id":"n-b6cbf920aaa38713","other":{"id":"n-b6cbf920aaa38713","name":"basis_map_A007318 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","first_index":0,"fingerprint":"f6-bc84b3931f09589a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","1"],["18","9","1"]],"card_rows":[["1"],["3","1"],["18","9","1"],["162","99","18","1"]],"row_sums_preview":["1","4","28","280","3640","58240","1106560","24344320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8229dbdd6a4b5cd9","steps":["basis A007318"],"edges":["e-b2419cddc44d1315"],"text":"A225470 \u2192 basis A007318"},"headline":"A225470 \u2192 basis A007318","chain":null}},{"id":"l-685c82da4aa305ee","kind":"oeis_match","node_a_id":"n-1a0f876fa1441737","node_b_id":null,"fingerprint":null,"oeis_id":"A007559","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":"A007559","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A007559","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-1a0f876fa1441737","oeis_id":"A007559","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":20,"oeis_terms_available":20,"term_count":200,"raw_size":43221}],"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: 0,         1;\n  2: 0,         3,         1;\n  3: 0,        18,         9,        1;\n  4: 0,       162,        99,       18,        1;\n  5: 0,      1944,      1350,      315,       30,       1;\n  6: 0,     29160,     22194,     6075,      765,      45,      1;\n  7: 0,    524880,    428652,   131544,    19845,    1575,     63,    1;\n  8: 0,  11022480,   9526572,  3191076,   548289,   52920,   2898,   84,   1;\n  9: 0, 264539520, 239660208, 86112396, 16350012, 1818369, 122472, 4914, 108, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-3+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n - 3 + 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; for(n = 1, N, my(i = n + 1); v[i] = (3*n - 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-1a0f876fa1441737","steps":[],"edges":[],"text":"basis-map coefficients of A371080, shifted"},"headline":"basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 1","chain":null}
