{"id":"n-2e4f18459aae0e07","revision":0,"first_index":0,"fingerprint":"f6-7d9acbbdb3cf6a7d","name":"basis_map_A013609 of basis_map_A013609 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"],["4","4"],["28","48","16"],["160","416","320","64"],["976","3328","3904","1792","256"],["5824","24768","39424","28672","9216","1024"],["35008","177920","356864","356352","183296","45056","4096"],["209920","1242112","3008512","3827712","2727936","1073152","212992","16384"],["1259776","8503296","24131584","37330944","34123776","18677760","5914624","983040","65536"],["7558144","57316352","186515456","339918848","379797504","267730944","118095872","31195136","4456448","262144"],["45349888","381693952","1400958976","2939813888","3885645824","3360227328","1913126912","703594496","159121408","19922944","1048576"],["272097280","2516901888","10285858816","24426364928","37298110464","38273024000","26794262528","12730761216","4003463168","790626304","88080384","4194304"],["1632587776","16461922304","74129653760","196542005248","340562673664","404777598976","336990306304","197153259520","80103866368","21961375744","3846176768","385875968","16777216"],["9795518464","106934419456","526067040256","1540376756224","2986817880064","4037407145984","3900971155456","2718051598336","1360762241024","481803894784","116937195520","18387828736","1677721600","67108864"],["58773127168","690584223744","3684952440832","11810353709056","25340202713088","38403234070528","42273838465024","34233775030272","20430953906176","8915462914048","2792500494336","607469436928","86637543424","7247757312","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","92","960","10256","108928","1158592","12318720","130990336","1392846848","14810504192","157483745280","1674564079616","17806057504768","189336254267392"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["12","16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199923,"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-c7a2083391c64372"]},{"id":199924,"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-c7a2083391c64372"]},{"id":199925,"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-c7a2083391c64372"]},{"id":199926,"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-c7a2083391c64372"]}],"region_exact":{},"incoming_edges":[{"id":"e-c7a2083391c64372","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-2e4f18459aae0e07","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":"e713db37cd26481cfeb24212f007bd506e998641db5fabe481ba41684241b2e9","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-bf2f08bd0370af8d","role":"sequence","name":"basis_map_A013609 of 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:       4,        4;\n  2:      28,       48,        16;\n  3:     160,      416,       320,        64;\n  4:     976,     3328,      3904,      1792,       256;\n  5:    5824,    24768,     39424,     28672,      9216,      1024;\n  6:   35008,   177920,    356864,    356352,    183296,     45056,      4096;\n  7:  209920,  1242112,   3008512,   3827712,   2727936,   1073152,    212992,    16384;\n  8: 1259776,  8503296,  24131584,  37330944,  34123776,  18677760,   5914624,   983040,   65536;\n  9: 7558144, 57316352, 186515456, 339918848, 379797504, 267730944, 118095872, 31195136, 4456448, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12+16*x)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(12 + 16*x)*P[n - 2, x] + (4 + 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] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (12 + 16*x)*v[i - 2] + (4 + 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 A013609","basis A013609"],"edges":["e-175e767cc5e2f6ad","e-c7a2083391c64372"],"text":"A081579 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A081579 \u2192 basis A013609 \u2192 basis A013609","chain":null}