{"id":"n-811ccbfd604dd9f0","revision":0,"first_index":0,"fingerprint":"f6-921c6d767cd7fcd1","name":"basis_map_A008277 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","1"],["4","3"],["8","11","2"],["16","31","8"],["32","91","42","4"],["64","247","152","20"],["128","675","586","136","8"],["256","1791","1992","600","48"],["512","4747","6858","2860","400","16"],["1024","12423","22392","11644","2080","112"],["2048","32435","73034","48240","11800","1104","32"],["4096","84175","232232","186176","55888","6608","256"],["8192","218011","735306","718164","270368","43456","2912","64"],["16384","562903","2297048","2672036","1201600","234528","19712","576"],["32768","1451331","7144906","9887960","5341608","1296736","147392","7424","128"]],"degrees":[1,1,2,2,3,3,4,4,5,5,6,6,7,7,8],"row_sums":["3","7","21","55","169","483","1533","4687","15393","49675","168693","569431","1996473","7004787","25310253"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","1"],["4","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196084,"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-863932fbf11b91ae"]},{"id":196085,"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-863932fbf11b91ae"]},{"id":196086,"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-863932fbf11b91ae"]}],"region_exact":{},"incoming_edges":[{"id":"e-863932fbf11b91ae","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-811ccbfd604dd9f0","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"54010072572df4b7460188361f662f219ae8e1f8b36eab563e94bf0d4d7109fc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"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,     1;\n  1:    4,     3;\n  2:    8,    11,     2;\n  3:   16,    31,     8;\n  4:   32,    91,    42,     4;\n  5:   64,   247,   152,    20;\n  6:  128,   675,   586,   136,    8;\n  7:  256,  1791,  1992,   600,   48;\n  8:  512,  4747,  6858,  2860,  400,  16;\n  9: 1024, 12423, 22392, 11644, 2080, 112;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) + 2*x*d/dx P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 4 + 3*x.","mathematica":"P[0, x_] := 2 + x;\nP[1, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] + 2*x*D[P[n - 2, x], 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 + x; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] + 2*x*deriv(v[i - 2], 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-ff9680a2cfd23e93","steps":["basis A008277"],"edges":["e-863932fbf11b91ae"],"text":"A199478 \u2192 basis A008277"},"headline":"A199478 \u2192 basis A008277","chain":null}