{"id":"n-692225cc4bc27507","revision":1,"first_index":0,"fingerprint":"f6-16a8138f8e411269","name":"basis_map_A013609 of A133080 * A007318.","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":true,"rows":[["1"],["3","2"],["4","8","4"],["12","32","28","8"],["16","64","96","64","16"],["48","224","416","384","176","32"],["64","384","960","1280","960","384","64"],["192","1280","3648","5760","5440","3072","960","128"],["256","2048","7168","14336","17920","14336","7168","2048","256"],["768","6656","25600","57344","82432","78848","50176","20480","4864","512"],["1024","10240","46080","122880","215040","258048","215040","122880","46080","10240","1024"],["3072","32768","158720","460800","890880","1204224","1161216","798720","384000","122880","23552","2048"],["4096","49152","270336","901120","2027520","3244032","3784704","3244032","2027520","901120","270336","49152","4096"],["12288","155648","909312","3244032","7884800","13787136","17842176","17301504","12570624","6758400","2613248","688128","110592","8192"],["16384","229376","1490944","5963776","16400384","32800768","49201152","56229888","49201152","32800768","16400384","5963776","1490944","229376","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","16","80","256","1280","4096","20480","65536","327680","1048576","5242880","16777216","83886080","268435456"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","8","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166313,"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-af466358510fd7f5"]},{"id":166314,"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-af466358510fd7f5"]},{"id":166315,"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-af466358510fd7f5"]}],"region_exact":{},"incoming_edges":[{"id":"e-af466358510fd7f5","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-692225cc4bc27507","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":"4b3308578ef272aca76f2d8e3813562aa4f78f34dca43b7a7899ae23e98b621c","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-d75669a8cd7f9393","role":"sequence","name":"A133080 * A007318.","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,    2;\n  2:   4,    8,     4;\n  3:  12,   32,    28,     8;\n  4:  16,   64,    96,    64,    16;\n  5:  48,  224,   416,   384,   176,    32;\n  6:  64,  384,   960,  1280,   960,   384,    64;\n  7: 192, 1280,  3648,  5760,  5440,  3072,   960,   128;\n  8: 256, 2048,  7168, 14336, 17920, 14336,  7168,  2048,  256;\n  9: 768, 6656, 25600, 57344, 82432, 78848, 50176, 20480, 4864, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+8*x+4*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(4 + 8*x + 4*x^2)*P[n - 2, 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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 8*x + 4*x^2)*v[i - 2]); 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-d75669a8cd7f9393","steps":["basis A013609"],"edges":["e-af466358510fd7f5"],"text":"A133087 \u2192 basis A013609"},"headline":"A133087 \u2192 basis A013609","chain":null}
