{"id":"n-2a67a442e762ff53","revision":0,"first_index":0,"fingerprint":"f6-23e3cf183ada7515","name":"basis_map_A013610 of Expansion of 1/sqrt(1-4*x*y-4*x^2*y).","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","6"],["8","42","54"],["32","252","648","540"],["136","1416","5454","9180","5670"],["592","7680","39420","99900","124740","61236"],["2624","40752","261360","885060","1666980","1653372","673596"],["11776","213024","1639008","6947640","17508960","26209008","21555072","7505784"],["53344","1101504","9885456","50334480","158947110","318549672","395400852","277714008","84440070"],["243392","5648832","57928608","344373120","1307286540","3284760276","5460169176","5786959464","3546482940","956987460"],["1116928","28781760","332009280","2256971040","10009364400","30248563212","63058689540","89506474200","82751268600","44978410620","10909657044"],["5149696","145884288","1869626880","14304755520","72580717440","256353124104","642941993232","1144669588920","1417242134880","1161782776440","567302166288","124965162504"],["23835904","736263936","10378539072","88263267840","504258456240","2038379633136","5976621291564","12803569003368","19884454324020","21826969987680","16069924825812","7123014262728","1437099368796"],["110690816","3702506496","56931997824","532753752960","3384320940000","15408794104704","51720313468824","129553802399736","242124469998840","333374237590680","328620689479368","219563790519528","89100160865352","16581915793800"],["515483648","18562242048","309191081472","3157173452160","22074996303360","111785352430752","422706650157792","1212338187228168","2649556125576360","4390239305972520","5428056223614024","4855021528442904","2969047295932536","1110988358184600","191876454185400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","104","1472","21856","333568","5183744","81590272","1296426496","20750839808","334081306624","5404163080192","87763693060096","1430025994108928","23367175920287744"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","6"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","-12"],["4","12"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","-6"],["4","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":202254,"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-1645e47027924141"]},{"id":202255,"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-1645e47027924141"]},{"id":202256,"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-1645e47027924141"]},{"id":202257,"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-1645e47027924141"]}],"region_exact":{},"incoming_edges":[{"id":"e-1645e47027924141","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-2a67a442e762ff53","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":"cc0e07cbdb4f271ea0d3681bf2aef9c14f4c7f1fe1c55a69e54406f2175ccad3","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-eacb751f39dbd056","role":"sequence","name":"Expansion of 1/sqrt(1-4*x*y-4*x^2*y).","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,       6;\n  2:      8,      42,       54;\n  3:     32,     252,      648,       540;\n  4:    136,    1416,     5454,      9180,       5670;\n  5:    592,    7680,    39420,     99900,     124740,      61236;\n  6:   2624,   40752,   261360,    885060,    1666980,    1653372,     673596;\n  7:  11776,  213024,  1639008,   6947640,   17508960,   26209008,   21555072,    7505784;\n  8:  53344, 1101504,  9885456,  50334480,  158947110,  318549672,  395400852,  277714008,   84440070;\n  9: 243392, 5648832, 57928608, 344373120, 1307286540, 3284760276, 5460169176, 5786959464, 3546482940, 956987460;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (4*n-4+(12*n-12)*x)*P(n-2,x) + (4*n-2+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (12*n - 12)*x)*P[n - 2, x] + (4*n - 2 + (12*n - 6)*x)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (12*n - 12)*x)*v[i - 2] + (4*n - 2 + (12*n - 6)*x)*v[i - 1])/(n)); 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-eacb751f39dbd056","steps":["basis A013610"],"edges":["e-1645e47027924141"],"text":"A115951 \u2192 basis A013610"},"headline":"A115951 \u2192 basis A013610","chain":null}