{"id":"n-f17d61ee7b0cc64e","revision":1,"first_index":1,"fingerprint":"f6-cfb345b8c9d5b7ff","name":"basis_map_A013610 of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","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","3"],["5","15","9"],["12","54","72","27"],["29","174","360","297","81"],["70","525","1476","1917","1134","243"],["169","1521","5409","9639","8991","4131","729"],["408","4284","18432","41850","53784","38880","14580","2187"],["985","11820","59688","164862","271026","270216","158922","50301","6561"],["2378","32103","186120","605934","1215972","1553742","1259712","623295","170586","19683"],["5741","86115","563697","2114154","5015358","7838694","8150220","5552793","2368521","570807","59049"],["13860","228690","1668168","7083693","19406952","35925120","45737460","39958677","23435892","8778618","1889568","177147"],["33461","602298","4844448","22977135","71422803","152955864","230916582","247133187","185774715","95541282","31886460","6200145","531441"],["80782","1575249","13850028","72575811","252435042","614326113","1074213576","1364187177","1256260914","827493003","378622188","113905521","20194758","1594323"],["195025","4095525","39077145","224212185","862951725","2353181625","4681189665","6890898015","7532257635","6076634175","3558233691","1465537131","401237955","65367243","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","29","165","941","5365","30589","174405","994381","5669525","32325149","184303845","1050819821","5991314485","34159851709"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167039,"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-79afa45c031edfc6"]},{"id":167040,"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-79afa45c031edfc6"]},{"id":167041,"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-79afa45c031edfc6"]},{"id":167042,"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-79afa45c031edfc6"]}],"region_exact":{},"incoming_edges":[{"id":"e-79afa45c031edfc6","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-f17d61ee7b0cc64e","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":"0aee107685adbd4d24b5ccfd1d7513f4283dea623c12428f64b34ac243377367","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-811c78acf96cbc19","role":"sequence","name":"Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","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-eb9a78b9fca420c1","kind":"fingerprint_related","node_a_id":"n-f17d61ee7b0cc64e","node_b_id":"n-3594f9d1943ddb79","fingerprint":"f6-cfb345b8c9d5b7ff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cfb345b8c9d5b7ff"},"other_id":"n-3594f9d1943ddb79","other":{"id":"n-3594f9d1943ddb79","name":"basis_map_A013610 of delannoy","first_index":0,"fingerprint":"f6-cfb345b8c9d5b7ff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["5","15","9"]],"card_rows":[["1"],["2","3"],["5","15","9"],["12","54","72","27"]],"row_sums_preview":["1","5","29","165","941","5365","30589","174405"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e00a1cb428776eaa","steps":["basis A013610"],"edges":["e-470609786340cbfa"],"text":"delannoy \u2192 basis A013610"},"headline":"delannoy \u2192 basis A013610","chain":null}},{"id":"l-0e10e1343541b169","kind":"oeis_match","node_a_id":"n-f17d61ee7b0cc64e","node_b_id":null,"fingerprint":null,"oeis_id":"A015537","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A015537","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A015537","relation":"match_row_sums"}],"bfiles":[],"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,     3;\n   3:    5,    15,      9;\n   4:   12,    54,     72,     27;\n   5:   29,   174,    360,    297,      81;\n   6:   70,   525,   1476,   1917,    1134,     243;\n   7:  169,  1521,   5409,   9639,    8991,    4131,     729;\n   8:  408,  4284,  18432,  41850,   53784,   38880,   14580,   2187;\n   9:  985, 11820,  59688, 164862,  271026,  270216,  158922,  50301,   6561;\n  10: 2378, 32103, 186120, 605934, 1215972, 1553742, 1259712, 623295, 170586, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*x)*P(n-2,x) + (2+3*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 3*x)*P[n - 2, x] + (2 + 3*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 + 3*x; for(n = 3, N, my(i = n); v[i] = (1 + 3*x)*v[i - 2] + (2 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-811c78acf96cbc19","steps":["basis A013610"],"edges":["e-79afa45c031edfc6"],"text":"A008288 \u2192 basis A013610"},"headline":"A008288 \u2192 basis A013610","chain":null}