{"id":"n-ca8f77e380f6014f","revision":1,"first_index":0,"fingerprint":"f6-60d64fb2105f3f8c","name":"basis_map_A013610 of Riordan array (1-x,x/(1-x)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1","6","9"],["2","15","36","27"],["4","36","117","162","81"],["8","84","342","675","648","243"],["16","192","936","2376","3321","2430","729"],["32","432","2448","7560","13770","14823","8748","2187"],["64","960","6192","22464","50220","70956","61965","30618","6561"],["128","2112","15264","63504","167832","292572","336798","247131","104976","19683"],["256","4608","36864","172800","526176","1088640","1551312","1504656","951345","354294","59049"],["512","9984","87552","456192","1570752","3755808","6368544","7663248","6416658","3562623","1180980","177147"],["1024","21504","205056","1175040","4510080","12223872","24004512","34432128","35823060","26375220","13049829","3897234","531441"],["2048","46080","474624","2965248","12545280","37977984","84680640","140877792","174942504","160219620","105225318","46943955","12754584","1594323"],["4096","98304","1087488","7354368","33986304","113591808","283295232","535797504","772518384","845266752","691109496","409563864","166341033","41452398","4782969"],["8192","208896","2469888","17971200","90035712","329142528","907365888","1921480704","3152429280","4008088656","3918019248","2892456216","1561373658","581927895","133923132","14348907"],["16384","442368","5566464","43352064","233985024","928392192","2802159360","6565059072","12069300672","17473465152","19860304464","17538970176","11800115964","5847976764","2013629949","430467210","43046721"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["16","80","400","2000","10000","50000","250000","1250000","6250000","31250000","156250000","781250000","3906250000","19531250000","97656250000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","6","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166989,"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-02e979b2d9fb24b7"]},{"id":166990,"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-02e979b2d9fb24b7"]},{"id":166991,"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-02e979b2d9fb24b7"]}],"region_exact":{},"incoming_edges":[{"id":"e-02e979b2d9fb24b7","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-ca8f77e380f6014f","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":"101b8c7fca352fba9d5f1e8cc75516c486ce4c60f48bb6b865b564198bd7475b","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-7717563017cb9e5f","role":"sequence","name":"Riordan array (1-x,x/(1-x)).","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-0e9bda781501ab26","kind":"oeis_match","node_a_id":"n-ca8f77e380f6014f","node_b_id":null,"fingerprint":null,"oeis_id":"A055842","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A055842","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A055842","relation":"match_row_sums"}],"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,    6,     9;\n  1:   2,   15,    36,     27;\n  2:   4,   36,   117,    162,      81;\n  3:   8,   84,   342,    675,     648,     243;\n  4:  16,  192,   936,   2376,    3321,    2430,     729;\n  5:  32,  432,  2448,   7560,   13770,   14823,    8748,    2187;\n  6:  64,  960,  6192,  22464,   50220,   70956,   61965,   30618,    6561;\n  7: 128, 2112, 15264,  63504,  167832,  292572,  336798,  247131,  104976,   19683;\n  8: 256, 4608, 36864, 172800,  526176, 1088640, 1551312, 1504656,  951345,  354294,   59049;\n  9: 512, 9984, 87552, 456192, 1570752, 3755808, 6368544, 7663248, 6416658, 3562623, 1180980, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 6*x + 9*x^2.","mathematica":"P[0, x_] := 1 + 6*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + 3*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 + 6*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 3*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-7717563017cb9e5f","steps":["basis A013610"],"edges":["e-02e979b2d9fb24b7"],"text":"A159854 \u2192 basis A013610"},"headline":"A159854 \u2192 basis A013610","chain":null}