{"id":"n-a6cd3f3657ee08fd","revision":1,"first_index":0,"fingerprint":"f6-dd0a803b420669cd","name":"basis_map_A013610 of Triangle of coefficients in expansion of (3+5x)^n.","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"],["8","15"],["64","240","225"],["512","2880","5400","3375"],["4096","30720","86400","108000","50625"],["32768","307200","1152000","2160000","2025000","759375"],["262144","2949120","13824000","34560000","48600000","36450000","11390625"],["2097152","27525120","154828800","483840000","907200000","1020600000","637875000","170859375"],["16777216","251658240","1651507200","6193152000","14515200000","21772800000","20412000000","10935000000","2562890625"],["134217728","2264924160","16986931200","74317824000","209018880000","391910400000","489888000000","393660000000","184528125000","38443359375"],["1073741824","20132659200","169869312000","849346560000","2786918400000","6270566400000","9797760000000","10497600000000","7381125000000","3075468750000","576650390625"],["8589934592","177167400960","1660944384000","9342812160000","35035545600000","91968307200000","172440576000000","230947200000000","216513000000000","135320625000000","50745234375000","8649755859375"],["68719476736","1546188226560","15945066086400","99656663040000","420426547200000","1261279641600000","2759049216000000","4434186240000000","5196312000000000","4330260000000000","2435771250000000","830376562500000","129746337890625"],["549755813888","13400297963520","150753352089600","1036429295616000","4858262323200000","16396635340800000","40991588352000000","76859228160000000","108083289600000000","112586760000000000","84440070000000000","43179581250000000","13493619140625000","1946195068359375"],["4398046511104","115448720916480","1407031286169600","10552734646272000","54412538019840000","204047017574400000","573882236928000000","1229747650560000000","2017554739200000000","2521943424000000000","2364321960000000000","1612037700000000000","755642671875000000","217973847656250000","29192926025390625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","23","529","12167","279841","6436343","148035889","3404825447","78310985281","1801152661463","41426511213649","952809757913927","21914624432020321","504036361936467383","11592836324538749809"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","15"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167438,"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-c78a98c6ca91439c"]},{"id":167439,"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-c78a98c6ca91439c"]},{"id":167440,"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-c78a98c6ca91439c"]}],"region_exact":{},"incoming_edges":[{"id":"e-c78a98c6ca91439c","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-a6cd3f3657ee08fd","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":"29313a737b116195b8ecac739fa9b33981005a11fdb0fbfe09935f4d8a6aaaaf","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-c44f7114310405ad","role":"sequence","name":"Triangle of coefficients in expansion of (3+5x)^n.","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":[{"id":"l-516c6dd8d2a233ec","kind":"oeis_match","node_a_id":"n-a6cd3f3657ee08fd","node_b_id":null,"fingerprint":null,"oeis_id":"A009967","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A009967","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009967","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-a6cd3f3657ee08fd","oeis_id":"A009967","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":28090}],"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:         8,         15;\n  2:        64,        240,         225;\n  3:       512,       2880,        5400,        3375;\n  4:      4096,      30720,       86400,      108000,        50625;\n  5:     32768,     307200,     1152000,     2160000,      2025000,       759375;\n  6:    262144,    2949120,    13824000,    34560000,     48600000,     36450000,     11390625;\n  7:   2097152,   27525120,   154828800,   483840000,    907200000,   1020600000,    637875000,    170859375;\n  8:  16777216,  251658240,  1651507200,  6193152000,  14515200000,  21772800000,  20412000000,  10935000000,   2562890625;\n  9: 134217728, 2264924160, 16986931200, 74317824000, 209018880000, 391910400000, 489888000000, 393660000000, 184528125000, 38443359375;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(8 + 15*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; for(n = 1, N, my(i = n + 1); v[i] = (8 + 15*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-c44f7114310405ad","steps":["basis A013610"],"edges":["e-c78a98c6ca91439c"],"text":"A013622 \u2192 basis A013610"},"headline":"A013622 \u2192 basis A013610","chain":null}