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DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["6","1"],["20","6"],["68","29","1"],["232","130","10"],["792","559","69","1"],["2704","2338","406","14"],["9232","9585","2183","126","1"],["31520","38706","11070","924","18"],["107616","154471","53865","6005","200","1"],["367424","610698","254166","36014","1760","22"],["1254464","2395937","1171135","203926","13445","291","1"],["4283008","9340474","5295238","1105884","93314","2990","26"],["14623104","36220423","23576889","5798597","604072","26278","399","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","7","26","98","372","1421","5462","21127","82238","322158","1270084","5039199","20120934","80849763"],"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"]],"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":[["4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196384,"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-dc26f38baa89ca6a"]},{"id":196385,"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-dc26f38baa89ca6a"]},{"id":196386,"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-dc26f38baa89ca6a"]}],"region_exact":{},"incoming_edges":[{"id":"e-dc26f38baa89ca6a","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-34bf3ddff9651c72","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":"6dc0f9cdaeef64eb8a1bdf571b5e3e5a44359f514aadca5da528b9a9049248ae","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-048e7c9e10af99dd","role":"sequence","name":"Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) 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;\n   3:     6,     1;\n   4:    20,     6;\n   5:    68,    29,     1;\n   6:   232,   130,    10;\n   7:   792,   559,    69,   1;\n   8:  2704,  2338,   406,  14;\n   9:  9232,  9585,  2183, 126,  1;\n  10: 31520, 38706, 11070, 924, 18;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2-x)*P(n-2,x) + x*d/dx P(n-2,x) + 4*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[-(2 - x)*P[n - 2, x] + x*D[P[n - 2, x], x] + 4*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; for(n = 3, N, my(i = n); v[i] = -(2 - x)*v[i - 2] + x*deriv(v[i - 2], x) + 4*v[i - 1]); 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-048e7c9e10af99dd","steps":["basis A008277"],"edges":["e-dc26f38baa89ca6a"],"text":"A201730 \u2192 basis A008277"},"headline":"A201730 \u2192 basis A008277","chain":null}