{"id":"n-f72d27cff21cd6ce","revision":2,"first_index":1,"fingerprint":"f6-e49d6044c973fe25","name":"basis_map_A007318 of Coefficients of a special case of Poisson-Charlier polynomials.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["5","5","1"],["16","25","11","1"],["65","144","107","28","1"],["326","961","1054","511","94","1"],["1957","7313","10844","7971","2904","421","1"],["13700","62525","118131","118196","66009","19496","2379","1"],["109601","593736","1371390","1750008","1331819","604136","151150","16080","1"],["986410","6204369","17002476","26506726","25704006","15870051","6089684","1327110","125682","1"],["9864101","70795717","225054759","415820030","491982301","386450583","201465865","67192406","13003803","1112093","1"],["108505112","876281825","3175433755","6798282239","9520608602","9111364417","6033208873","2728668097","806466242","140602199","10976195","1"],["1302061345","11697961960","47654407481","116178896252","188311850374","213046453280","171638245701","98448869828","39390679030","10467990336","1662409033","119481308","1"],["16926797486","167578760337","758797180010","2077706665893","3831180274492","5011294034462","4767220776746","3322703968585","1683740678476","604844258734","146173838810","21333400661","1421542654","1"],["236975164805","2564729469409","12787479239170","38887346757273","80467384785215","119629896706178","131447869099403","108069616814589","66470919100299","30204956542762","9854569310871","2185574987025","295272917614","18348340141","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","11","53","345","2947","31411","400437","5927921","99816515","1882741659","39310397557","899919305929","22410922177347","603120939234755"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","4","2"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161799,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-1db18e82b98a79ad"]},{"id":161800,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-1db18e82b98a79ad","e-21bb7128247ad5fa"]},{"id":161801,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-1db18e82b98a79ad","e-21bb7128247ad5fa"]},{"id":161802,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-1db18e82b98a79ad","e-21bb7128247ad5fa"]},{"id":161803,"position":4,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-21bb7128247ad5fa"]}],"region_exact":{},"incoming_edges":[{"id":"e-1db18e82b98a79ad","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-f72d27cff21cd6ce","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":"e41fc1030407d873a896ed21161d52095b7b297fb22c3033ae69546424690b95","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c9dce8c25621fdea","role":"sequence","name":"Coefficients of a special case of Poisson-Charlier polynomials.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-21bb7128247ad5fa","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-f72d27cff21cd6ce","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":"675c87e1d4e907b2b4bc3de30719468cbd58755dcaae19b454f37ddd211a5728","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c9dce8c25621fdea","role":"sequence","name":"Coefficients of a special case of Poisson-Charlier polynomials.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-209fed6f99014465","kind":"fingerprint_related","node_a_id":"n-f72d27cff21cd6ce","node_b_id":"n-51eaee9ee85e261b","fingerprint":"f6-e49d6044c973fe25","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e49d6044c973fe25"},"other_id":"n-51eaee9ee85e261b","other":{"id":"n-51eaee9ee85e261b","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","first_index":1,"fingerprint":"f6-e49d6044c973fe25","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","5"]],"card_rows":[["1"],["1","2"],["1","5","5"],["1","11","25","16"]],"row_sums_preview":["1","3","11","53","345","2947","31411","400437"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3272d5bc697b9cb4","steps":["x \u21a6 x/(x+1)"],"edges":["e-d35851ec1bbd1c60"],"text":"A094816 \u2192 x \u21a6 x/(x+1)"},"headline":"A094816 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-5b3c7ee1f13810cc","kind":"oeis_match","node_a_id":"n-f72d27cff21cd6ce","node_b_id":null,"fingerprint":null,"oeis_id":"A081367","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A081367","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A081367","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-f72d27cff21cd6ce","oeis_id":"A081367","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":18,"oeis_terms_available":18,"term_count":200,"raw_size":39898}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      2,       1;\n   3:      5,       5,        1;\n   4:     16,      25,       11,        1;\n   5:     65,     144,      107,       28,        1;\n   6:    326,     961,     1054,      511,       94,        1;\n   7:   1957,    7313,    10844,     7971,     2904,      421,       1;\n   8:  13700,   62525,   118131,   118196,    66009,    19496,    2379,       1;\n   9: 109601,  593736,  1371390,  1750008,  1331819,   604136,  151150,   16080,      1;\n  10: 986410, 6204369, 17002476, 26506726, 25704006, 15870051, 6089684, 1327110, 125682, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) + (n+(n-1)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-(n - 2 + (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x] + (n + (n - 1)*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = -(n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 2] + (n + (n - 1)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c9dce8c25621fdea","steps":["basis A007318"],"edges":["e-1db18e82b98a79ad"],"text":"A046716 \u2192 basis A007318"},"headline":"A046716 \u2192 basis A007318","chain":null}