{"id":"n-c8021aa0b636ccac","revision":0,"first_index":0,"fingerprint":"f6-61e4d58d94cdaa75","name":"basis_map_A013611 of Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2","4"],["4","12"],["8","36","32"],["16","92","128"],["32","228","480","256"],["64","540","1472","1280"],["128","1252","4256","5632","2048"],["256","2844","11520","19968","12288"],["512","6372","30048","65280","61440","16384"],["1024","14108","75840","197376","245760","114688"],["2048","30948","186912","568320","890880","638976","131072"],["4096","67356","451456","1569792","2961408","2834432","1048576"],["8192","145636","1072864","4201728","9289728","11239424","6422528","1048576"],["16384","313116","2514624","10952960","27770880","40599552","31195136","9437184"],["32768","669924","5825440","27939328","79964160","137396224","133955584","62914560","8388608"]],"degrees":[1,1,2,2,3,3,4,4,5,5,6,6,7,7,8],"row_sums":["6","16","76","236","996","3356","13316","46876","180036","648796","2449156","8937116","33428676","122799836","457086596"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","4"],["4","12"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204440,"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-7f723efd807c2db6"]},{"id":204441,"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-7f723efd807c2db6"]},{"id":204442,"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-7f723efd807c2db6"]},{"id":204443,"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-7f723efd807c2db6"]}],"region_exact":{},"incoming_edges":[{"id":"e-7f723efd807c2db6","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-c8021aa0b636ccac","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":"8beda78528b188e5d246384dfb3c83bbeae13a3b9dc6df2e8bf38d8265473106","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-ff9680a2cfd23e93","role":"sequence","name":"Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.","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:    2,     4;\n  1:    4,    12;\n  2:    8,    36,    32;\n  3:   16,    92,   128;\n  4:   32,   228,   480,    256;\n  5:   64,   540,  1472,   1280;\n  6:  128,  1252,  4256,   5632,   2048;\n  7:  256,  2844, 11520,  19968,  12288;\n  8:  512,  6372, 30048,  65280,  61440,  16384;\n  9: 1024, 14108, 75840, 197376, 245760, 114688;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+8*x)*P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 2 + 4*x, P(1,x) = 4 + 12*x.","mathematica":"P[0, x_] := 2 + 4*x;\nP[1, x_] := 4 + 12*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 8*x)*P[n - 2, 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] = 2 + 4*x; v[2] = 4 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 8*x)*v[i - 2] + 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-ff9680a2cfd23e93","steps":["basis A013611"],"edges":["e-7f723efd807c2db6"],"text":"A199478 \u2192 basis A013611"},"headline":"A199478 \u2192 basis A013611","chain":null}
