{"id":"n-b6cbf920aaa38713","revision":1,"first_index":0,"fingerprint":"f6-bc84b3931f09589a","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.","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"],["3","1"],["18","9","1"],["162","99","18","1"],["1944","1350","315","30","1"],["29160","22194","6075","765","45","1"],["524880","428652","131544","19845","1575","63","1"],["11022480","9526572","3191076","548289","52920","2898","84","1"],["264539520","239660208","86112396","16350012","1818369","122472","4914","108","1"],["7142567040","6735365136","2564694900","527562720","65445975","5125113","255150","7830","135","1"],["214277011200","209203521120","83676212136","18391576500","2490941970","219199365","12779613","490050","11880","165","1"],["7071141369600","7117993208160","2970518521608","690598236636","100592661510","9724521015","640926594","28951263","882090","17325","198","1"],["254561089305600","263318896863360","114056659986048","27832055040504","4311934050996","450675418050","32797878399","1683172062","60706503","1505790","24453","234","1"],["9927882482918400","10523998066976640","4711528636319232","1199506806565704","195997483029348","21888275354946","1729792675611","98441588817","4050725679","119432313","2459457","33579","273","1"],["416971064282572800","451935801295937280","208408200792384384","55090814512078800","9431401093798320","1115305047937080","94539567730608","5864339405925","268572067335","9066882825","222729507","3869775","45045","315","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","28","280","3640","58240","1106560","24344320","608608000","17041024000","528271744000","17961239296000","664565853952000","26582634158080000","1143053268797440000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161965,"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-b2419cddc44d1315"]},{"id":161966,"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-b2419cddc44d1315"]},{"id":161967,"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-b2419cddc44d1315"]}],"region_exact":{},"incoming_edges":[{"id":"e-b2419cddc44d1315","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-b6cbf920aaa38713","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":"dc017776ba6c448a48c444a7dc3bf72b140b764f6fddf8240641b6bd0c9adefa","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-8229dbdd6a4b5cd9","role":"sequence","name":"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.","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-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-1a0f876fa1441737","other":{"id":"n-1a0f876fa1441737","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","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"],["0","1"],["0","3","1"]],"card_rows":[["1"],["0","1"],["0","3","1"],["0","18","9","1"]],"row_sums_preview":["1","1","4","28","280","3640","58240","1106560"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-4da80552b92b8615","kind":"oeis_match","node_a_id":"n-b6cbf920aaa38713","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":1,"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":[],"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,          1;\n  2:         18,          9,          1;\n  3:        162,         99,         18,         1;\n  4:       1944,       1350,        315,        30,        1;\n  5:      29160,      22194,       6075,       765,       45,       1;\n  6:     524880,     428652,     131544,     19845,     1575,      63,      1;\n  7:   11022480,    9526572,    3191076,    548289,    52920,    2898,     84,    1;\n  8:  264539520,  239660208,   86112396,  16350012,  1818369,  122472,   4914,  108,   1;\n  9: 7142567040, 6735365136, 2564694900, 527562720, 65445975, 5125113, 255150, 7830, 135, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+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 + 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 + 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-8229dbdd6a4b5cd9","steps":["basis A007318"],"edges":["e-b2419cddc44d1315"],"text":"A225470 \u2192 basis A007318"},"headline":"A225470 \u2192 basis A007318","chain":null}
