{"id":"n-53575d46e8908019","revision":0,"first_index":0,"fingerprint":"f6-8a820a157fa8f06f","name":"basis_map_A013609 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","2"],["5","6"],["17","24","4"],["73","112","28"],["382","618","192","8"],["2365","3966","1404","104"],["16937","29144","11256","1120","16"],["137861","241848","99384","11872","336"],["1257686","2239650","964000","130480","5280","32"],["12714721","22914070","10223080","1515440","76880","992"],["141119617","256809792","117897180","18728320","1111920","21504","64"],["1706150125","3130061016","1470817380","246701440","16450800","412800","2752"],["22321071242","41229842234","19752142704","3461641400","252428960","7611744","78848","128"],["314201147513","583760152542","284260937268","51651315800","4043859120","139878816","1932224","7296"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["1","4","11","45","213","1200","7839","58473","491301","4597128","47445183","535688397","6570596313","87024817260","1238059230579"],"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","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0"],["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199691,"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-80189eee586fd712"]},{"id":199692,"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-80189eee586fd712"]},{"id":199693,"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-80189eee586fd712"]},{"id":199694,"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-80189eee586fd712"]}],"region_exact":{},"incoming_edges":[{"id":"e-80189eee586fd712","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-53575d46e8908019","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":"10f5a0d2192317cd58b3ed7f7273fe448056e096dd3b89a0887e0274203dba56","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-bf8bba6616804a77","role":"sequence","name":"Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:       5,       6;\n  3:      17,      24,      4;\n  4:      73,     112,     28;\n  5:     382,     618,    192,      8;\n  6:    2365,    3966,   1404,    104;\n  7:   16937,   29144,  11256,   1120,   16;\n  8:  137861,  241848,  99384,  11872,  336;\n  9: 1257686, 2239650, 964000, 130480, 5280, 32;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x)*P(n-2,x) + n*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[(1 + 2*x)*P[n - 2, x] + n*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; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 2] + n*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-bf8bba6616804a77","steps":["basis A013609"],"edges":["e-80189eee586fd712"],"text":"A249247 \u2192 basis A013609"},"headline":"A249247 \u2192 basis A013609","chain":null}