{"id":"n-543627493a90d10d","revision":1,"first_index":0,"fingerprint":"f6-beeca1c718627537","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","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"],["8","1"],["64","17","1"],["512","217","27","1"],["4096","2465","487","38","1"],["32768","26281","7335","905","50","1"],["262144","269297","99631","17290","1505","63","1"],["2097152","2685817","1265607","289821","35350","2324","77","1"],["16777216","26269505","15341887","4453638","714021","65562","3402","92","1"],["134217728","253202761","179688375","64331905","13021890","1566327","113190","4782","108","1"],["1073741824","2413042577","2050086511","887339330","220594585","33384141","3150987","184920","6510","125","1"],["8589934592","22791125017","22913907687","11810819141","3534474350","654588418","77497959","5924787","289080","8635","143","1"],["68719476736","213710059745","251930201887","152832918238","54224511341","12044123784","1739559844","166369764","10550067","435875","11209","162","1"],["549755813888","1992110014441","2733012078615","1933092302505","803527054330","210798120533","36397961600","4235106304","335170836","17959942","637637","14287","182","1"],["4398046511104","18478745943857","29322230800591","23997027406170","11575416954465","3543902621259","720369582933","99924556160","9597839680","640489850","29437408","909090","17927","203","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","82","757","7087","67340","649931","6376149","63625324","646147067","6681531511","70378569810","755378218653","8263282235101","92145933070698"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166037,"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-9a1b575c999b0b50"]},{"id":166038,"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-9a1b575c999b0b50"]},{"id":166039,"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-9a1b575c999b0b50"]}],"region_exact":{},"incoming_edges":[{"id":"e-9a1b575c999b0b50","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-543627493a90d10d","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":"93b1aca9030282a99b3ae5db958b8327938183644679c73bd40b57b4e422825b","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-f211cb6153c37c6d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","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:         8,         1;\n  2:        64,        17,         1;\n  3:       512,       217,        27,        1;\n  4:      4096,      2465,       487,       38,        1;\n  5:     32768,     26281,      7335,      905,       50,       1;\n  6:    262144,    269297,     99631,    17290,     1505,      63,      1;\n  7:   2097152,   2685817,   1265607,   289821,    35350,    2324,     77,    1;\n  8:  16777216,  26269505,  15341887,  4453638,   714021,   65562,   3402,   92,   1;\n  9: 134217728, 253202761, 179688375, 64331905, 13021890, 1566327, 113190, 4782, 108, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(8 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (8 + x)*v[i - 1] + x*deriv(v[i - 1], x)); 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-f211cb6153c37c6d","steps":["basis A008277"],"edges":["e-9a1b575c999b0b50"],"text":"A027466 \u2192 basis A008277"},"headline":"A027466 \u2192 basis A008277","chain":null}
