{"id":"n-cc7cbbeddaf74600","revision":1,"first_index":0,"fingerprint":"f6-7f70594cc517f6b8","name":"basis_map_A013611 of Pascal-(1,4,1) array.","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","4"],["8","32","16"],["24","144","224","64"],["80","640","1600","1280","256"],["256","2560","8960","12800","6656","1024"],["832","9984","44800","92160","86016","32768","4096"],["2688","37632","206336","558080","772096","520192","155648","16384"],["8704","139264","902144","3026944","5595136","5636096","2932736","720896","65536"],["28160","506880","3788800","15196160","35315712","48087040","37355520","15728640","3276800","262144"],["91136","1822720","15441920","72089600","202227712","349503488","368967680","230686720","81264640","14680064","1048576"],["294912","6488064","61440000","327352320","1077215232","2265317376","3054764032","2597847040","1350041600","407896064","65011712","4194304"],["954368","22904832","239763456","1435893760","5426184192","13473153024","22238724096","24240979968","17107517440","7574913024","2000683008","285212672","16777216"],["3088384","80297984","920715264","6123290624","26142441472","74947756032","146834194432","196704468992","178144673792","106812145664","41087401984","9630121984","1241513984","67108864"],["9994240","279838720","3488153600","25509232640","121457082368","395176837120","897984757760","1433529221120","1599392972800","1230222917632","638624727040","216761630720","45634027520","5368709120","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","56","456","3856","32256","270656","2269056","19027456","159545856","1337824256","11217862656","94063661056","788739219456","6613708537856"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167849,"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-8275de16895af39b"]},{"id":167850,"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-8275de16895af39b"]},{"id":167851,"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-8275de16895af39b"]},{"id":167852,"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-8275de16895af39b"]}],"region_exact":{},"incoming_edges":[{"id":"e-8275de16895af39b","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-cc7cbbeddaf74600","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":"adbac7fac11a6663ee7b76c1451ec879629befd8a05bea4c7e2a0ceb3b8df84e","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-ed9434bfd889c409","role":"sequence","name":"Pascal-(1,4,1) array.","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,      4;\n  2:     8,     32,      16;\n  3:    24,    144,     224,       64;\n  4:    80,    640,    1600,     1280,      256;\n  5:   256,   2560,    8960,    12800,     6656,     1024;\n  6:   832,   9984,   44800,    92160,    86016,    32768,     4096;\n  7:  2688,  37632,  206336,   558080,   772096,   520192,   155648,    16384;\n  8:  8704, 139264,  902144,  3026944,  5595136,  5636096,  2932736,   720896,   65536;\n  9: 28160, 506880, 3788800, 15196160, 35315712, 48087040, 37355520, 15728640, 3276800, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+16*x)*P(n-2,x) + (2+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(4 + 16*x)*P[n - 2, x] + (2 + 4*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; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 16*x)*v[i - 2] + (2 + 4*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-ed9434bfd889c409","steps":["basis A013611"],"edges":["e-8275de16895af39b"],"text":"A081579 \u2192 basis A013611"},"headline":"A081579 \u2192 basis A013611","chain":null}