{"id":"n-445f9a43ace6155b","revision":0,"first_index":0,"fingerprint":"f6-44562cdf8dd879d1","name":"basis_map_A008277 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","2"],["8","20","6"],["32","176","132","20"],["136","1488","2116","760","70"],["592","12432","30040","19240","4060","252"],["2624","103712","402144","409520","148680","20664","924"],["11776","867456","5217856","7944832","4417840","1031856","101640","3432"],["53344","7285248","66539344","145880224","116662840","40737984","6632472","487344","12870"],["243392","61456064","840376032","2587328672","2861522160","1401841392","335489616","40243632","2290860","48620"],["1116928","520636800","10558671616","44861041600","66825869920","44135694624","14564816112","2536179360","233444640","10599160","184756"],["5149696","4427874816","132323885056","766155165952","1508490364480","1307253127488","571433259936","135470553504","17926914720","1306127680","48406072","705432"],["23835904","37788879360","1656752587648","12951631176704","33239291556416","37065381800832","20880257290816","6472723390848","1154785343712","120032667040","7094260888","218683920","2704156"],["110690816","323492530688","20743808427264","217432092543744","719709044875392","1017728888944768","724316066714240","285580900761728","65964352918464","9172541069120","768594567312","37588708976","978904472","10400600"],["515483648","2776746681344","259884570519552","3633256290273792","15384176838541056","27278663623154944","24162617623464320","11875888788667904","3454076094049152","616979504261120","68725857616416","4740496455968","195012914096","4347450800","40116600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","34","360","4570","66616","1088268","19596688","384291670","8130840440","184248255516","4444841534832","113585984178244","3061778472057584","86741986349650712"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","-4"],["4","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-4"],["0","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","-2"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2"],["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196621,"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-7fec2eeeaaa65e45"]},{"id":196622,"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-7fec2eeeaaa65e45"]},{"id":196623,"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-7fec2eeeaaa65e45"]}],"region_exact":{},"incoming_edges":[{"id":"e-7fec2eeeaaa65e45","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-445f9a43ace6155b","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"c0689ae47cbda9a68df788dc4e7bf99659801619c50058e62a7624505f5a0815","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"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,        2;\n  2:      8,       20,         6;\n  3:     32,      176,       132,         20;\n  4:    136,     1488,      2116,        760,         70;\n  5:    592,    12432,     30040,      19240,       4060,        252;\n  6:   2624,   103712,    402144,     409520,     148680,      20664,       924;\n  7:  11776,   867456,   5217856,    7944832,    4417840,    1031856,    101640,     3432;\n  8:  53344,  7285248,  66539344,  145880224,  116662840,   40737984,   6632472,   487344,   12870;\n  9: 243392, 61456064, 840376032, 2587328672, 2861522160, 1401841392, 335489616, 40243632, 2290860, 48620;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (4*n-4+(4*n-4)*x)*P(n-2,x) + (4*n-4)*x*d/dx P(n-2,x) + (4*n-2+(4*n-2)*x)*P(n-1,x) + (4*n-2)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (4*n - 4)*x)*P[n - 2, x] + (4*n - 4)*x*D[P[n - 2, x], x] + (4*n - 2 + (4*n - 2)*x)*P[n - 1, x] + (4*n - 2)*x*D[P[n - 1, x], 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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (4*n - 4)*x)*v[i - 2] + (4*n - 4)*x*deriv(v[i - 2], x) + (4*n - 2 + (4*n - 2)*x)*v[i - 1] + (4*n - 2)*x*deriv(v[i - 1], x))/(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 A008277"],"edges":["e-7fec2eeeaaa65e45"],"text":"A115951 \u2192 basis A008277"},"headline":"A115951 \u2192 basis A008277","chain":null}
