{"id":"n-798a9ec526dde6cd","revision":1,"first_index":0,"fingerprint":"f6-82102477dd106c0e","name":"basis_map_A013611 of Triangle T(n,k) = T(n-1,k) + 3*T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1"],["4","12"],["7","24"],["10","36"],["22","120","144"],["43","276","432"],["73","504","864"],["139","1128","2736","1728"],["268","2472","7344","6912"],["487","4860","15984","17280"],["904","9912","37728","55296","20736"],["1708","20544","89424","164160","103680"],["3169","40968","195696","407808","311040"]],"degrees":[0,0,0,1,1,1,2,2,2,3,3,3,4,4,4],"row_sums":["1","1","1","16","31","46","286","751","1441","5731","16996","38611","124576","379516","958681"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","12"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167677,"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-3e5def6a97b3833e"]},{"id":167678,"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-3e5def6a97b3833e"]},{"id":167679,"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-3e5def6a97b3833e"]},{"id":167680,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-3e5def6a97b3833e"]}],"region_exact":{},"incoming_edges":[{"id":"e-3e5def6a97b3833e","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-798a9ec526dde6cd","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"1bc354b5d631381c443fef0d7ab58bcc5af03bfe13ae7b96cb089bb0fb027315","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a4ffad71e3e1c62c","role":"sequence","name":"Triangle T(n,k) = T(n-1,k) + 3*T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","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:   1;\n  2:   1;\n  3:   4,   12;\n  4:   7,   24;\n  5:  10,   36;\n  6:  22,  120,  144;\n  7:  43,  276,  432;\n  8:  73,  504,  864;\n  9: 139, 1128, 2736, 1728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+12*x)*P(n-3,x) + P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + 12*x)*P[n - 3, x] + 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] = 1; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (3 + 12*x)*v[i - 3] + 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-a4ffad71e3e1c62c","steps":["basis A013611"],"edges":["e-3e5def6a97b3833e"],"text":"A317496 \u2192 basis A013611"},"headline":"A317496 \u2192 basis A013611","chain":null}