{"id":"n-2fd9e2edf97805a2","revision":0,"first_index":0,"fingerprint":"f6-3bef121b79a49566","name":"basis_map_A013609 of basis_map_A013610 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["5","6"],["29","66","36"],["169","564","612","216"],["985","4332","7020","4968","1296"],["5741","31338","67500","71712","37584","7776"],["33461","218142","586008","830304","654480","272160","46656"],["195025","1478280","4757400","8426160","8871120","5552064","1912896","279936"],["1136689","9821688","36828072","78271056","103129200","86266944","44743104","13156992","1679616"],["6625109","64271982","275141160","682168608","1079530416","1130824800","784147392","347120640","89019648","10077696"],["38613965","415560858","1999650780","5665587552","10467064944","13175864352","11445090048","6774451200","2615162112","594584064","60466176"],["225058681","2660776956","14216128668","45309261384","95716384560","140726750976","146941579008","108970126848","56248660224","19248399360","3930301440","362797056"],["1311738121","16899452964","99261782964","351486752760","835686810720","1405482948864","1714564889856","1528695783936","988697560320","452387773440","139011738624","25758590976","2176782336"],["7645370045","106606369554","682751286900","2659181952960","7027324996320","13306291806528","18573345453312","19350593915904","15048111405312","8627263603200","3544466770944","988259180544","167612239872","13060694016"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","11","131","1561","18601","221651","2641211","31472881","375033361","4468927451","53252096051","634556225161","7561422605881","90102515045411"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199564,"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-f4c213ae3901f81b"]},{"id":199565,"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-f4c213ae3901f81b"]},{"id":199566,"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-f4c213ae3901f81b"]},{"id":199567,"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-f4c213ae3901f81b"]}],"region_exact":{},"incoming_edges":[{"id":"e-f4c213ae3901f81b","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-2fd9e2edf97805a2","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":"7d0b712cc6a7aa254158f225fa959abd6538d6a3939fba66d4d75f8375801a50","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ee90405a2d071e0a","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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-f6b7197920ddd9c4","kind":"fingerprint_related","node_a_id":"n-2fd9e2edf97805a2","node_b_id":"n-15fb5012236a46df","fingerprint":"f6-3bef121b79a49566","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3bef121b79a49566"},"other_id":"n-15fb5012236a46df","other":{"id":"n-15fb5012236a46df","name":"basis_map_A013610 of basis_map_A007318 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","first_index":0,"fingerprint":"f6-3bef121b79a49566","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["29","66","36"],["169","564","612","216"],["985","4332","7020","4968","1296"]],"card_rows":[["29","66","36"],["169","564","612","216"],["985","4332","7020","4968","1296"],["5741","31338","67500","71712","37584","7776"]],"row_sums_preview":["131","1561","18601","221651","2641211","31472881","375033361","4468927451"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8eb9bb3a09f0b7fd","steps":["basis A007318","basis A013610"],"edges":["e-1acefd6886502727","e-61378e2264964559"],"text":"A123519 \u2192 basis A007318 \u2192 basis A013610"},"headline":"A123519 \u2192 basis A007318 \u2192 basis A013610","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:       1;\n  2:       5,       6;\n  3:      29,      66,       36;\n  4:     169,     564,      612,      216;\n  5:     985,    4332,     7020,     4968,      1296;\n  6:    5741,   31338,    67500,    71712,     37584,     7776;\n  7:   33461,  218142,   586008,   830304,    654480,   272160,    46656;\n  8:  195025, 1478280,  4757400,  8426160,   8871120,  5552064,  1912896,   279936;\n  9: 1136689, 9821688, 36828072, 78271056, 103129200, 86266944, 44743104, 13156992, 1679616;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (6 + 6*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (6 + 6*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-0eba401b117f16a9","steps":["basis A013610","basis A013609"],"edges":["e-0b8f676aef1665e7","e-f4c213ae3901f81b"],"text":"A085478 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A085478 \u2192 basis A013610 \u2192 basis A013609","chain":null}