{"id":"n-279334a58393797d","revision":0,"first_index":1,"fingerprint":"f6-4139c2f8a6e84b16","name":"basis_map_A008277 of shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["8","7","1"],["28","41","13","1"],["100","221","117","20","1"],["356","1135","899","255","28","1"],["1268","5659","6319","2646","477","37","1"],["4516","27665","41985","24369","6408","808","47","1"],["16084","133421","268525","207748","74733","13608","1276","58","1"],["57284","637279","1671651","1678843","793696","195661","26276","1912","70","1"],["204020","3022747","10203055","13051974","7890861","2528973","455961","47194","2750","83","1"],["726628","14264129","61361905","98580765","74729020","30285968","7038491","971385","80056","3827","97","1"],["2587924","67055405","364908621","728309420","682081045","342610436","99809048","17633048","1926695","129643","5183","112","1"],["9217028","314328559","2151310259","5289030871","6049831620","3709695329","1327485764","291920484","40598638","3604564","202013","6861","128","1"],["32826932","1470285403","12597569743","37891951206","52458647901","38807748349","16798150025","4505197084","775405500","87206174","6422572","304706","8907","145","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","16","83","459","2674","16407","105799","715454","5062673","37407619","288042272","2307056581","19187232119","165431724648"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","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":196429,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-632851b77ec9a448"]},{"id":196430,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-632851b77ec9a448"]},{"id":196431,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-632851b77ec9a448"]}],"region_exact":{},"incoming_edges":[{"id":"e-632851b77ec9a448","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-279334a58393797d","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":"347b23670b9961795cbe0b622bb31368b260158f013175efb8e48d7f6ccd57a4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ec08416ae48317c3","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     2,      1;\n   3:     8,      7,       1;\n   4:    28,     41,      13,       1;\n   5:   100,    221,     117,      20,      1;\n   6:   356,   1135,     899,     255,     28,      1;\n   7:  1268,   5659,    6319,    2646,    477,     37,     1;\n   8:  4516,  27665,   41985,   24369,   6408,    808,    47,    1;\n   9: 16084, 133421,  268525,  207748,  74733,  13608,  1276,   58,  1;\n  10: 57284, 637279, 1671651, 1678843, 793696, 195661, 26276, 1912, 70, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-2,x) + x*d/dx P(n-2,x) + (3+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + (3 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (2 + x)*v[i - 2] + x*deriv(v[i - 2], x) + (3 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b0fc620949fb5990","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-3a1dbaeb4ca6a6b2","e-632851b77ec9a448"],"text":"A122542 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A122542 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}