{"id":"n-406d5bfd4a642364","revision":0,"first_index":0,"fingerprint":"f6-32632a63e62b0230","name":"basis_map_A013609 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","4"],["8","28","24"],["32","168","288","160"],["136","944","2424","2720","1120"],["592","5120","17520","29600","24640","8064"],["2624","27168","116160","262240","329280","217728","59136"],["11776","142016","728448","2058560","3458560","3451392","1892352","439296"],["53344","734336","4393536","14913920","31396960","41948928","34712832","16253952","3294720"],["243392","3765888","25746048","102036480","258229440","432561024","479356416","338697216","138378240","24893440"],["1116928","19187840","147559680","668732160","1977158400","3983349888","5536016640","5238604800","3228825600","1169991680","189190144"],["5149696","97256192","830945280","4238446080","14336931840","33758436096","56444838912","66994836480","55298580480","30220636160","9837887488","1444724736"],["23835904","490842624","4612684032","26152079360","99606608640","268428593664","524696519424","749362977792","775860433920","567769579520","278677082112","82349309952","11076222976"],["110690816","2468337664","25303110144","157852963840","668507840000","2029141610496","4540603651584","7582481347584","9447319664640","8671828971520","5698785517568","2538381361152","686725824512","85201715200"],["515483648","12374828032","137418258432","935458800640","4360493096960","14720704846848","37110048847872","70955321428992","103381552834560","114200199393280","94130799386624","56129000718336","22883476668416","5708514918400","657270374400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","60","648","7344","85536","1014336","12182400","147702528","1803907584","22159733760","273508669440","3389106769920","42134712606720","525323149885440"],"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":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","-8"],["4","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","-4"],["4","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199458,"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-7adabb58a59dfd52"]},{"id":199459,"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-7adabb58a59dfd52"]},{"id":199460,"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-7adabb58a59dfd52"]},{"id":199461,"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-7adabb58a59dfd52"]}],"region_exact":{},"incoming_edges":[{"id":"e-7adabb58a59dfd52","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-406d5bfd4a642364","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":"e9a563af3064c16a0ea21d175e4ba74f596876720d3ec690b7512a9fc3b140fe","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,       4;\n  2:      8,      28,       24;\n  3:     32,     168,      288,       160;\n  4:    136,     944,     2424,      2720,      1120;\n  5:    592,    5120,    17520,     29600,     24640,      8064;\n  6:   2624,   27168,   116160,    262240,    329280,    217728,     59136;\n  7:  11776,  142016,   728448,   2058560,   3458560,   3451392,   1892352,    439296;\n  8:  53344,  734336,  4393536,  14913920,  31396960,  41948928,  34712832,  16253952,   3294720;\n  9: 243392, 3765888, 25746048, 102036480, 258229440, 432561024, 479356416, 338697216, 138378240, 24893440;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (4*n-4+(8*n-8)*x)*P(n-2,x) + (4*n-2+(8*n-4)*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[Together[((4*n - 4 + (8*n - 8)*x)*P[n - 2, x] + (4*n - 2 + (8*n - 4)*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (8*n - 8)*x)*v[i - 2] + (4*n - 2 + (8*n - 4)*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 A013609"],"edges":["e-7adabb58a59dfd52"],"text":"A115951 \u2192 basis A013609"},"headline":"A115951 \u2192 basis A013609","chain":null}