{"id":"n-e34f9664814ccfa0","revision":0,"first_index":0,"fingerprint":"f6-9e2859b846aec920","name":"basis_map_A013609 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.","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"],["6","12","4"],["16","48","40","8"],["48","192","256","128","16"],["160","800","1472","1216","416","32"],["576","3456","8128","9472","5568","1408","64"],["2176","15232","43648","65920","55680","25216","4992","128"],["8448","67584","228352","423936","468480","309248","113664","18432","256"],["33280","299520","1165312","2566144","3505152","3046400","1648640","509952","70144","512"],["132096","1320960","5813248","14802944","24070144","25948160","18565120","8511488","2274304","272384","1024"],["526336","5789696","28424192","82126848","154816512","199127040","177106944","107630592","42805248","10074112","1071104","2048"],["2101248","25214976","136585216","441303040","945745920","1412530176","1501151232","1135706112","599715840","210616320","44310528","4243456","4096"],["8396800","109158400","646561792","2309210112","5542338560","9422561280","11640668160","10533371904","6931144704","3235471360","1017364480","193576960","16883712","8192"],["33570816","469991424","3021389824","11817123840","31389794304","59826798592","84203716608","88639930368","69808865280","40633663488","16994942976","4837801984","840351744","67338240","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","22","112","640","4096","28672","212992","1638400","12845056","101711872","809500672","6459228160","51606716416","412585295872"],"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"],["6","12","4"],["16","48","40","8"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["24","48","24"],["-8","-16","-8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-18","-18"],["6","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199103,"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-289c399f4e1cc521"]},{"id":199104,"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-289c399f4e1cc521"]},{"id":199105,"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-289c399f4e1cc521"]},{"id":199106,"position":3,"claim":"interlacing","from_index":3,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 3","region_exact":null,"edges":["e-289c399f4e1cc521"]}],"region_exact":{},"incoming_edges":[{"id":"e-289c399f4e1cc521","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-e34f9664814ccfa0","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":"6b68965b91009629dc1a8f1b607204a2e5e6b8f6ec6b483a2d236eed60ac8fa9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-02236a48f5253afe","role":"sequence","name":"Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.","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:     6,     12,       4;\n  3:    16,     48,      40,       8;\n  4:    48,    192,     256,     128,      16;\n  5:   160,    800,    1472,    1216,     416,      32;\n  6:   576,   3456,    8128,    9472,    5568,    1408,      64;\n  7:  2176,  15232,   43648,   65920,   55680,   25216,    4992,    128;\n  8:  8448,  67584,  228352,  423936,  468480,  309248,  113664,  18432,   256;\n  9: 33280, 299520, 1165312, 2566144, 3505152, 3046400, 1648640, 509952, 70144, 512;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = -(8*n-24+(16*n-48)*x+(8*n-24)*x^2)*P(n-2,x) + (6*n-18+(6*n-18)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 6 + 12*x + 4*x^2, P(3,x) = 16 + 48*x + 40*x^2 + 8*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[2, x_] := 6 + 12*x + 4*x^2;\nP[3, x_] := 16 + 48*x + 40*x^2 + 8*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 24 + (16*n - 48)*x + (8*n - 24)*x^2)*P[n - 2, x] + (6*n - 18 + (6*n - 18)*x)*P[n - 1, x])/(n - 3)]];\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; v[3] = 6 + 12*x + 4*x^2; v[4] = 16 + 48*x + 40*x^2 + 8*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(8*n - 24 + (16*n - 48)*x + (8*n - 24)*x^2)*v[i - 2] + (6*n - 18 + (6*n - 18)*x)*v[i - 1])/(n - 3)); 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-02236a48f5253afe","steps":["basis A013609"],"edges":["e-289c399f4e1cc521"],"text":"A146986 \u2192 basis A013609"},"headline":"A146986 \u2192 basis A013609","chain":null}
