{"id":"n-9116f92e53f934a9","revision":1,"first_index":0,"fingerprint":"f6-1258bebed215369f","name":"basis_map_A013609 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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"],["2","2"],["10","26","16"],["100","440","628","288"],["1700","10680","24756","24992","9216"],["44200","362680","1177656","1887592","1489216","460800"],["1635400","16601560","69686232","154632136","191007616","124273152","33177600"],["81770000","990347200","5111264480","14560857536","24704330128","24932403968","13837648896","3251404800"],["5315050000","74839128000","458996632800","1600697593280","3469571222928","4782760514304","4090794886144","1982560370688","416179814400"],["435834100000","6997846596000","49761662625600","205614657162560","543817850391456","954256900287264","1110252383981056","825278721951744","355301524832256","67421129932800"],["44019244100000","793949326196000","6425497244385600","30719412898538560","96048534322049056","205143517007304864","302986870839539456","305403627713337344","200941198398406656","77869839089664000","13484225986560000"],["5370347780200000","107514474868112000","976046400754475200","5302738706763019520","19152019108736316352","48271254380827064960","86609129358191590720","110582065324195704320","98422504111233249280","58127890381353418752","20489576630059008000","3263182688747520000"],["778700428129000000","17136259016573840000","172490896871415160000","1049998475897926688000","4304231518314515492800","12515113388535983528576","26460445018615975362880","40977828727167555253760","46118897909497183989760","36774225289331421511680","19711821041188340760576","6374159559325384704000","939796614359285760000"],["132379072781930000000","3176364777525154800000","35115508015742535120000","236801664045185861040000","1086618842966966854320000","3582399529241423436424320","8728383978489878244348288","15909861299910684065792640","21690718754397154954030080","21839805792596389845524480","15780697724796038400245760","7746202637006974576754688","2314231355493058609152000","317651255653438586880000"],["26078677338040210000000","677636457702972055600000","8162890071891140100240000","60415206959072688391920000","306890164370205327828720000","1131687293703611423869031040","3123792259225144001214946176","6555769195650437032745678976","10509737224181227679734640640","12805203492865173541548113920","11670001322482604384294010880","7712035427610421044517011456","3492415010738866580090781696","969755988717006376402944000","124519292216147926056960000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","52","1456","71344","5422144","591013696","87470027008","16881715212544","4119138511860736","1239860692070081536","451309291913509679104","195416923398549691052032","99271797086463243054432256","58471088483926850159060598784"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["0","0"],["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166829,"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-8c95d99ebe6d3ebc"]},{"id":166830,"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-8c95d99ebe6d3ebc"]},{"id":166831,"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-8c95d99ebe6d3ebc"]},{"id":166832,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-8c95d99ebe6d3ebc"]}],"region_exact":{},"incoming_edges":[{"id":"e-8c95d99ebe6d3ebc","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-9116f92e53f934a9","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":"0d9b011202ed7a576c04821a24218578c79902666b346a76b2299bd84f28e135","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e734a631f8279507","role":"sequence","name":"Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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-1dd902541977d20b","kind":"oeis_match","node_a_id":"n-9116f92e53f934a9","node_b_id":null,"fingerprint":null,"oeis_id":"A277353","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":"A277353","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A277353","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-9116f92e53f934a9","oeis_id":"A277353","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":16,"oeis_terms_available":16,"term_count":200,"raw_size":76665}],"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:            2,             2;\n  2:           10,            26,             16;\n  3:          100,           440,            628,             288;\n  4:         1700,         10680,          24756,           24992,            9216;\n  5:        44200,        362680,        1177656,         1887592,         1489216,          460800;\n  6:      1635400,      16601560,       69686232,       154632136,       191007616,       124273152,         33177600;\n  7:     81770000,     990347200,     5111264480,     14560857536,     24704330128,     24932403968,      13837648896,      3251404800;\n  8:   5315050000,   74839128000,   458996632800,   1600697593280,   3469571222928,   4782760514304,    4090794886144,   1982560370688,    416179814400;\n  9: 435834100000, 6997846596000, 49761662625600, 205614657162560, 543817850391456, 954256900287264, 1110252383981056, 825278721951744, 355301524832256, 67421129932800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2+1+2*n^2*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[(n^2 + 1 + 2*n^2*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] = (n^2 + 1 + 2*n^2*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-e734a631f8279507","steps":["basis A013609"],"edges":["e-8c95d99ebe6d3ebc"],"text":"A008955 \u2192 basis A013609"},"headline":"A008955 \u2192 basis A013609","chain":null}