{"id":"n-2cb7ff9fe3f3bf65","revision":0,"first_index":0,"fingerprint":"f6-23551e2957101112","name":"basis_map_A008277 of Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3"],["6","1"],["12","5"],["24","17","1"],["48","51","7"],["96","143","33","1"],["192","385","131","9"],["384","1009","471","54","1"],["768","2595","1589","266","11"],["1536","6583","5129","1165","80","1"],["3072","16529","16031","4717","470","13"],["6144","41177","48903","18058","2425","111","1"],["12288","101955","146397","66298","11447","757","15"],["24576","251231","431777","235673","50652","4494","147","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","3","7","17","42","106","273","717","1919","5229","14494","40832","116819","339157","998551"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196072,"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-588c86d28b8a12bb"]},{"id":196073,"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-588c86d28b8a12bb"]},{"id":196074,"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-588c86d28b8a12bb"]}],"region_exact":{},"incoming_edges":[{"id":"e-588c86d28b8a12bb","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-2cb7ff9fe3f3bf65","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":"e0d519f7e772b8c4712cf843520cbc0acd18153f040368039eca00d782f1841e","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-d4c5acf24bc40277","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.","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:   3;\n  2:   6,    1;\n  3:  12,    5;\n  4:  24,   17,    1;\n  5:  48,   51,    7;\n  6:  96,  143,   33,   1;\n  7: 192,  385,  131,   9;\n  8: 384, 1009,  471,  54,  1;\n  9: 768, 2595, 1589, 266, 11;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-2,x) + x*d/dx P(n-2,x) + 2*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 2, x] + x*D[P[n - 2, x], x] + 2*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 2] + x*deriv(v[i - 2], x) + 2*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-d4c5acf24bc40277","steps":["basis A008277"],"edges":["e-588c86d28b8a12bb"],"text":"A208509 \u2192 basis A008277"},"headline":"A208509 \u2192 basis A008277","chain":null}