{"id":"n-2c8e7f457b5a8224","revision":0,"first_index":0,"fingerprint":"f6-f19985dacafc7a88","name":"basis_map_A013611 of Number triangle T(n,k)=C(n,k)C(2,n-k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","4"],["6","24","16"],["10","72","144","64"],["15","160","576","768","256"],["21","300","1600","3840","3840","1024"],["28","504","3600","12800","23040","18432","4096"],["36","784","7056","33600","89600","129024","86016","16384"],["45","1152","12544","75264","268800","573440","688128","393216","65536"],["55","1620","20736","150528","677376","1935360","3440640","3538944","1769472","262144"],["66","2200","32400","276480","1505280","5419008","12902400","19660800","17694720","7864320","1048576"],["78","2904","48400","475200","3041280","13246464","39739392","81100800","108134400","86507520","34603008","4194304"],["91","3744","69696","774400","5702400","29196288","105971712","272498688","486604800","576716800","415236096","150994944","16777216"],["105","4732","97344","1208064","10067200","59304960","253034496","787218432","1771241472","2811494400","2998927360","1962934272","654311424","67108864"],["120","5880","132496","1817088","16912896","112752640","553512960","2024275968","5510529024","11021058048","15744368640","15267266560","9160359936","2818572288","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","46","290","1775","10625","62500","362500","2078125","11796875","66406250","371093750","2060546875","11376953125","62500000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","4"],["6","24","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","12","48","64"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-3","-24","-48"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204187,"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-c3edb2d8cccc84d9"]},{"id":204188,"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-c3edb2d8cccc84d9"]},{"id":204189,"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-c3edb2d8cccc84d9"]}],"region_exact":{},"incoming_edges":[{"id":"e-c3edb2d8cccc84d9","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-2c8e7f457b5a8224","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":"85b90fc156bbc73ad0d0d9a3df69f0c40aac4f7da8830a4d53893310358a8458","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-1815d3e1c9c57d5b","role":"sequence","name":"Number triangle T(n,k)=C(n,k)C(2,n-k).","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,    4;\n  2:  6,   24,    16;\n  3: 10,   72,   144,     64;\n  4: 15,  160,   576,    768,    256;\n  5: 21,  300,  1600,   3840,   3840,    1024;\n  6: 28,  504,  3600,  12800,  23040,   18432,    4096;\n  7: 36,  784,  7056,  33600,  89600,  129024,   86016,   16384;\n  8: 45, 1152, 12544,  75264, 268800,  573440,  688128,  393216,   65536;\n  9: 55, 1620, 20736, 150528, 677376, 1935360, 3440640, 3538944, 1769472, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+12*x+48*x^2+64*x^3)*P(n-3,x) - (3+24*x+48*x^2)*P(n-2,x) + (3+12*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 4*x, P(2,x) = 6 + 24*x + 16*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[2, x_] := 6 + 24*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + 12*x + 48*x^2 + 64*x^3)*P[n - 3, x] - (3 + 24*x + 48*x^2)*P[n - 2, x] + (3 + 12*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] = 3 + 4*x; v[3] = 6 + 24*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = (1 + 12*x + 48*x^2 + 64*x^3)*v[i - 3] - (3 + 24*x + 48*x^2)*v[i - 2] + (3 + 12*x)*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-1815d3e1c9c57d5b","steps":["basis A013611"],"edges":["e-c3edb2d8cccc84d9"],"text":"A106246 \u2192 basis A013611"},"headline":"A106246 \u2192 basis A013611","chain":null}