{"id":"n-a8289b7270086fb2","revision":0,"first_index":0,"fingerprint":"f6-9bfdbf18d27088d9","name":"basis_map_A013609 of Riordan array (1, x/(1-3*x+2*x^2)).","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"],["1","2"],["4","10","4"],["14","44","36","8"],["48","184","224","104","16"],["164","744","1192","848","272","32"],["560","2936","5808","5568","2752","672","64"],["1912","11376","26720","32192","21600","8128","1600","128"],["6528","43456","118016","171072","145280","74368","22528","3712","256"],["22288","164128","505536","855936","880064","571776","235648","59648","8448","512"],["76096","614176","2114368","4092672","4941568","3898496","2041088","702464","152576","18944","1024"],["259808","2280640","8674752","18887552","26191488","24333568","15490048","6772736","1998336","379904","41984","2048"],["887040","8413824","35031552","84714368","132657920","141920256","106545152","56666112","21233664","5478400","925696","92160","4096"],["3028544","30868096","139604352","371145472","647677440","784329728","679041024","426209280","194270208","63621120","14575616","2215936","200704","8192"],["10340096","112701824","550090496","1594361856","3067684864","4148833280","4071733248","2949586944","1587032064","632068096","183693312","37830656","5226496","434176","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","18","102","576","3252","18360","103656","585216","3303984","18653472","105312864","594570240","3356795712","18951633792"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","2"],["4","10","4"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["4"],["-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-8","-4"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199149,"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-e8ca8860a1636b06"]},{"id":199150,"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-e8ca8860a1636b06"]},{"id":199151,"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-e8ca8860a1636b06"]},{"id":199152,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-e8ca8860a1636b06"]}],"region_exact":{},"incoming_edges":[{"id":"e-e8ca8860a1636b06","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-a8289b7270086fb2","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":"ec82272e2103b7dae1c8d606ec96a5f297d2ba91281a6a3178525b278f616e14","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-d7abbe24afd98bff","role":"sequence","name":"Riordan array (1, x/(1-3*x+2*x^2)).","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:     1,      2;\n  2:     4,     10,      4;\n  3:    14,     44,     36,      8;\n  4:    48,    184,    224,    104,     16;\n  5:   164,    744,   1192,    848,    272,     32;\n  6:   560,   2936,   5808,   5568,   2752,    672,     64;\n  7:  1912,  11376,  26720,  32192,  21600,   8128,   1600,   128;\n  8:  6528,  43456, 118016, 171072, 145280,  74368,  22528,  3712,  256;\n  9: 22288, 164128, 505536, 855936, 880064, 571776, 235648, 59648, 8448, 512;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(2*n-4)*P(n-2,x) + (4*n-8+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 4 + 10*x + 4*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 4 + 10*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4)*P[n - 2, x] + (4*n - 8 + (2*n - 4)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 4 + 10*x + 4*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(2*n - 4)*v[i - 2] + (4*n - 8 + (2*n - 4)*x)*v[i - 1])/(n - 2)); 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-d7abbe24afd98bff","steps":["basis A013609"],"edges":["e-e8ca8860a1636b06"],"text":"A206306 \u2192 basis A013609"},"headline":"A206306 \u2192 basis A013609","chain":null}
