{"id":"n-4d70fdb92495e233","revision":0,"first_index":0,"fingerprint":"f6-e3ba5e0ca89c8da7","name":"basis_map_A013609 of Triangle T(n,k) read by rows; related to number of preorders.","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":false,"rows":[["1"],["2","4"],["9","30","24"],["44","224","368","192"],["265","1800","4500","4880","1920"],["1854","15852","53496","88848","72384","23040"],["14833","153230","653100","1468040","1832320","1201536","322560"],["133496","1619648","8358080","23760640","40147840","40271872","22173696","5160960"],["1334961","18625824","113012424","389259360","831937680","1128903552","948987648","451381248","92897280"],["14684570","231823860","1618614480","6557610080","16981255200","29133464640","33095121920","23988648960","10059816960","1857945600"],["176214841","3107376206","24557788992","114511355728","348776249360","724776293472","1040257305472","1017810742784","649360521216","243797114880","40874803200"],["2290792932","44651986656","394257643248","2081061273024","7294745234112","17825097206784","30974063323392","38262050958336","32916229459968","18773948841984","6386219089920","980995276800"],["32071101049","685013919304","6686344053532","39431645412016","156450667632688","439879914073600","898482879181888","1343010538465024","1457629073984512","1119936036950016","578013615489024","179857379819520","25505877196800"],["481066515734","11177455714908","119559069121800","779434741100400","3454791567316320","10992741718195776","25828137510924672","45362610642074880","59543266752076800","57677208297533440","40065477286281216","18894932619558912","5420556783452160","714164561510400"],["7697064251745","193331173016670","2249555286656340","16070358192946200","78734625119607600","279826591532510880","743877954819679680","1502382929411295360","2316182072807116800","2711984039799091200","2373566693561057280","1505426577043783680","653934639891087360","174094348242124800","21424936845312000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","63","828","13365","255474","5645619","141626232","3976339977","123538986270","4208005766151","155935611087156","6245601057278973","268864540071377418","12379956350789537595"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"],["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"],["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200029,"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-c5d5b472de8f6fb7"]},{"id":200030,"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-c5d5b472de8f6fb7"]},{"id":200031,"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-c5d5b472de8f6fb7"]},{"id":200032,"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-c5d5b472de8f6fb7"]}],"region_exact":{},"incoming_edges":[{"id":"e-c5d5b472de8f6fb7","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-4d70fdb92495e233","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":"fbec80af1a3aeb871394dd3a8b1c597e4d1f4c8acecd30d3aeadd3ab2e6a74ba","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-f364085508d20047","role":"sequence","name":"Triangle T(n,k) read by rows; related to number of preorders.","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":[],"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:        2,         4;\n  2:        9,        30,         24;\n  3:       44,       224,        368,        192;\n  4:      265,      1800,       4500,       4880,        1920;\n  5:     1854,     15852,      53496,      88848,       72384,       23040;\n  6:    14833,    153230,     653100,    1468040,     1832320,     1201536,      322560;\n  7:   133496,   1619648,    8358080,   23760640,    40147840,    40271872,    22173696,     5160960;\n  8:  1334961,  18625824,  113012424,  389259360,   831937680,  1128903552,   948987648,   451381248,    92897280;\n  9: 14684570, 231823860, 1618614480, 6557610080, 16981255200, 29133464640, 33095121920, 23988648960, 10059816960, 1857945600;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+1+(2*n+2)*x)*P(n-2,x) + (n+1+(2*n+2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(n + 1 + (2*n + 2)*x)*P[n - 2, x] + (n + 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; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (2*n + 2)*x)*v[i - 2] + (n + 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-f364085508d20047","steps":["basis A013609"],"edges":["e-c5d5b472de8f6fb7"],"text":"A079510 \u2192 basis A013609"},"headline":"A079510 \u2192 basis A013609","chain":null}
