{"id":"n-d6a51c5da97a7289","revision":0,"first_index":0,"fingerprint":"f6-19181af399122da8","name":"basis_map_A013609 of Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is the 1-tiered binomial coefficient.","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":true,"rows":[["1"],["4","6"],["16","44","28"],["64","248","304","120"],["256","1264","2256","1744","496"],["1024","6112","14208","16192","9088","2016"],["4096","28608","81600","122240","101760","44736","8128"],["16384","130944","441600","817280","898560","587904","212224","32640"],["65536","589568","2291968","5040896","6872320","5954816","3205888","981248","130816"],["262144","2620928","11530240","29345792","47681536","51344384","36671488","16762880","4452352","523776"],["1048576","11533312","56613888","163540992","308195328","396232704","352192512","213823488","84897792","19913728","2096128"],["4194304","50329600","272609280","880711680","1887191040","2818142208","2994216960","2264494080","1195130880","419338240","88059904","8386560"],["16777216","218099712","1291800576","4613509120","11072286720","18822684672","23251329024","21036736512","13839851520","6458552320","2029817856","385830912","33550336"],["67108864","939515904","6039699456","23621779456","62744985600","119583940608","168302542848","177159831552","139512840192","81197670400","33955282944","9663135744","1677623296","134209536"],["268435456","4026515456","27917074432","118647193600","345471746048","729327419392","1151567020032","1381877612544","1266718801920","882863128576","460623527936","174478360576","45364314112","7247544320","536854528"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","88","736","6016","48640","391168","3137536","25133056","201195520","1610088448","12882804736","103070826496","824600166400","6596935548928"],"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","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-8","-16","-8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199391,"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-2b2554b17e246e10"]},{"id":199392,"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-2b2554b17e246e10"]},{"id":199393,"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-2b2554b17e246e10"]},{"id":199394,"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-2b2554b17e246e10"]}],"region_exact":{},"incoming_edges":[{"id":"e-2b2554b17e246e10","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-d6a51c5da97a7289","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":"79bf9f2ca47ce419e9b75fc34960f87545fe6555e5e6e5563998ab2e8e85731a","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-13e23eca5f474574","role":"sequence","name":"Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is the 1-tiered binomial coefficient.","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,       6;\n  2:     16,      44,       28;\n  3:     64,     248,      304,      120;\n  4:    256,    1264,     2256,     1744,      496;\n  5:   1024,    6112,    14208,    16192,     9088,     2016;\n  6:   4096,   28608,    81600,   122240,   101760,    44736,     8128;\n  7:  16384,  130944,   441600,   817280,   898560,   587904,   212224,    32640;\n  8:  65536,  589568,  2291968,  5040896,  6872320,  5954816,  3205888,   981248,  130816;\n  9: 262144, 2620928, 11530240, 29345792, 47681536, 51344384, 36671488, 16762880, 4452352, 523776;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(8+16*x+8*x^2)*P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-(8 + 16*x + 8*x^2)*P[n - 2, x] + (6 + 6*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 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 16*x + 8*x^2)*v[i - 2] + (6 + 6*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-13e23eca5f474574","steps":["basis A013609"],"edges":["e-2b2554b17e246e10"],"text":"A308737 \u2192 basis A013609"},"headline":"A308737 \u2192 basis A013609","chain":null}