{"id":"n-e739b38b5baf2db5","revision":1,"first_index":1,"fingerprint":"f6-c365b714cf5f4c2a","name":"basis_map_A008277 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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"],["3","1"],["8","7","1"],["21","35","12","1"],["55","154","94","18","1"],["144","636","612","201","25","1"],["377","2534","3602","1800","375","33","1"],["987","9877","19932","14201","4400","638","42","1"],["2584","37961","105935","103338","44626","9471","1015","52","1"],["6765","144551","547704","711762","411320","119756","18564","1534","63","1"],["17711","547008","2777136","4714938","3546376","1359897","285817","33852","2226","75","1"],["46368","2061192","13884984","30355002","29128250","14305796","3913686","622803","58275","3125","88","1"],["121393","7744128","68708976","191300058","230706376","142214721","49243153","10107864","1261602","95700","4268","102","1"],["317811","29036713","337404024","1186154322","1777116440","1354118348","581489412","149698990","23927211","2406877","151096","5695","117","1"],["832040","108720535","1647347857","7263029898","13395263026","12467848503","6538279903","2068371448","411636709","52714035","4366857","230724","7449","133","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","16","69","322","1619","8722","50078","304983","1962020","13285037","94379570","701508342","5441827057","43958649118"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"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":162231,"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-13742d707ef3ea03"]},{"id":162232,"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-13742d707ef3ea03"]},{"id":162233,"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-13742d707ef3ea03"]}],"region_exact":{},"incoming_edges":[{"id":"e-13742d707ef3ea03","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-e739b38b5baf2db5","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":"f61c0ab010e2aeb263067079cec7f61e9c75479b32b4b0bd77f2bec4dc3f3d30","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-976cdb6e4a693ec8","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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:    3,      1;\n   3:    8,      7,      1;\n   4:   21,     35,     12,      1;\n   5:   55,    154,     94,     18,      1;\n   6:  144,    636,    612,    201,     25,      1;\n   7:  377,   2534,   3602,   1800,    375,     33,     1;\n   8:  987,   9877,  19932,  14201,   4400,    638,    42,    1;\n   9: 2584,  37961, 105935, 103338,  44626,   9471,  1015,   52,  1;\n  10: 6765, 144551, 547704, 711762, 411320, 119756, 18564, 1534, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -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) = 3 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, 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] = 3 + x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (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-976cdb6e4a693ec8","steps":["basis A008277"],"edges":["e-13742d707ef3ea03"],"text":"A078812 \u2192 basis A008277"},"headline":"A078812 \u2192 basis A008277","chain":null}