{"id":"n-009cfc21b5a01b60","revision":1,"first_index":0,"fingerprint":"f6-8df9886dc486d5ab","name":"basis_map_A013609 of Pascal-(1,8,1) array.","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"],["2","2"],["12","24","4"],["40","120","88","8"],["176","704","832","256","16"],["672","3360","5696","3648","672","32"],["2752","16512","36032","34048","12864","1664","64"],["10880","76160","204416","260480","157568","40064","3968","128"],["43776","350208","1113600","1778688","1483776","614400","115200","9216","256"],["174592","1571328","5781504","11139072","11953152","7037952","2131968","313344","20992","512"],["699392","6993920","29217792","65888256","86513664","66637824","29091840","6807552","818176","47104","1024"],["2795520","30750720","143816704","371828736","578654208","553857024","321122304","108417024","20432896","2070528","104448","2048"],["11186176","134234112","694779904","2025881600","3647287296","4182343680","3048898560","1379008512","373166080","58474496","5111808","229376","4096"],["44736512","581574656","3300573184","10717020160","21924831232","29348585472","25893175296","14861107200","5402484736","1206771712","161136640","12369920","499712","8192"],["178962432","2505474048","15470280704","55358717952","126876106752","194362179584","201792208896","141323010048","65576648704","19656966144","3712303104","430637056","29442048","1081344","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","40","256","1984","14080","103936","753664","5509120","40124416","292716544","2133852160","15560605696","113454874624","827274035200"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166560,"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-0ac257a9997732d3"]},{"id":166561,"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-0ac257a9997732d3"]},{"id":166562,"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-0ac257a9997732d3"]},{"id":166563,"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-0ac257a9997732d3"]}],"region_exact":{},"incoming_edges":[{"id":"e-0ac257a9997732d3","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-009cfc21b5a01b60","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":"84e5b8c000a232ef648451a608c3cd56f2e758515f8dc313b53af1065abf18e0","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-1c472d1762258215","role":"sequence","name":"Pascal-(1,8,1) array.","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:      2,       2;\n  2:     12,      24,       4;\n  3:     40,     120,      88,        8;\n  4:    176,     704,     832,      256,       16;\n  5:    672,    3360,    5696,     3648,      672,      32;\n  6:   2752,   16512,   36032,    34048,    12864,    1664,      64;\n  7:  10880,   76160,  204416,   260480,   157568,   40064,    3968,    128;\n  8:  43776,  350208, 1113600,  1778688,  1483776,  614400,  115200,   9216,   256;\n  9: 174592, 1571328, 5781504, 11139072, 11953152, 7037952, 2131968, 313344, 20992, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+16*x)*P(n-2,x) + (2+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(8 + 16*x)*P[n - 2, x] + (2 + 2*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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 16*x)*v[i - 2] + (2 + 2*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-1c472d1762258215","steps":["basis A013609"],"edges":["e-0ac257a9997732d3"],"text":"A143683 \u2192 basis A013609"},"headline":"A143683 \u2192 basis A013609","chain":null}