{"id":"n-9e17ee3310d09339","revision":1,"first_index":0,"fingerprint":"f6-594cd89c06cea858","name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["5","1"],["20","9","1"],["80","56","13","1"],["320","304","108","17","1"],["1280","1536","736","176","21","1"],["5120","7424","4480","1440","260","25","1"],["20480","34816","25344","10240","2480","360","29","1"],["81920","159744","136192","66304","20160","3920","476","33","1"],["327680","720896","704512","401408","146944","35840","5824","608","37","1"],["1310720","3211264","3538944","2310144","989184","290304","59136","8256","756","41","1"],["5242880","14155776","17367040","12779520","6266880","2150400","526848","92160","11280","920","45","1"],["20971520","61865984","83623936","68485120","37847040","14868480","4257792","895488","137280","14960","1100","49","1"],["83886080","268435456","396361728","357564416","219873280","97320960","31899648","7839744","1444608","197120","19360","1296","53","1"],["335544320","1157627904","1853882368","1826619392","1237057536","609157120","224919552","63258624","13618176","2233088","274560","24544","1508","57","1"],["1342177280","4966055936","8573157376","9160359936","6774849536","3673686016","1508835328","477954048","117731328","22550528","3331328","372736","30576","1736","61","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["6","30","150","750","3750","18750","93750","468750","2343750","11718750","58593750","292968750","1464843750","7324218750","36621093750"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["5","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162745,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-7d4164eb4f384618"]}],"region_exact":{},"incoming_edges":[{"id":"e-7d4164eb4f384618","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-9e17ee3310d09339","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"6bfcc0eb07adfb02fe19f9cb6da3bc4e3b4896cf7d8d029d9d477d69510f71ec","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-3dcaebc50e901d48","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-3dcaebc50e901d48","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-2b29547e49f6d821","kind":"oeis_match","node_a_id":"n-9e17ee3310d09339","node_b_id":null,"fingerprint":null,"oeis_id":"A003948","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A003948","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-2d4eed88d9ee6684","kind":"oeis_match","node_a_id":"n-9e17ee3310d09339","node_b_id":null,"fingerprint":null,"oeis_id":"A168923","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168923","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-4cc19aac54cb5733","kind":"oeis_match","node_a_id":"n-9e17ee3310d09339","node_b_id":null,"fingerprint":null,"oeis_id":"A168779","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168779","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-4ccf83ed6557e181","kind":"oeis_match","node_a_id":"n-9e17ee3310d09339","node_b_id":null,"fingerprint":null,"oeis_id":"A168683","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168683","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-624dbee3e04c9fa8","kind":"oeis_match","node_a_id":"n-9e17ee3310d09339","node_b_id":null,"fingerprint":null,"oeis_id":"A168875","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168875","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-8afc03cb5beac179","kind":"oeis_match","node_a_id":"n-9e17ee3310d09339","node_b_id":null,"fingerprint":null,"oeis_id":"A167897","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A167897","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-bd3f84687c939747","kind":"oeis_match","node_a_id":"n-9e17ee3310d09339","node_b_id":null,"fingerprint":null,"oeis_id":"A168731","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168731","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fc9d5e9fab2e7ef5","kind":"oeis_match","node_a_id":"n-9e17ee3310d09339","node_b_id":null,"fingerprint":null,"oeis_id":"A168827","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168827","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A003948","relation":"match_row_sums"},{"oeis_id":"A167897","relation":"match_row_sums"},{"oeis_id":"A168683","relation":"match_row_sums"},{"oeis_id":"A168731","relation":"match_row_sums"},{"oeis_id":"A168779","relation":"match_row_sums"},{"oeis_id":"A168827","relation":"match_row_sums"},{"oeis_id":"A168875","relation":"match_row_sums"},{"oeis_id":"A168923","relation":"match_row_sums"}],"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:       5,       1;\n  1:      20,       9,       1;\n  2:      80,      56,      13,       1;\n  3:     320,     304,     108,      17,      1;\n  4:    1280,    1536,     736,     176,     21,      1;\n  5:    5120,    7424,    4480,    1440,    260,     25,     1;\n  6:   20480,   34816,   25344,   10240,   2480,    360,    29,    1;\n  7:   81920,  159744,  136192,   66304,  20160,   3920,   476,   33,   1;\n  8:  327680,  720896,  704512,  401408, 146944,  35840,  5824,  608,  37,  1;\n  9: 1310720, 3211264, 3538944, 2310144, 989184, 290304, 59136, 8256, 756, 41, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) for n >= 1, with P(0,x) = 5 + x.","mathematica":"P[0, x_] := 5 + x;\nP[n_, x_] := P[n, x] = Expand[(4 + 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] = 5 + x; for(n = 1, N, my(i = n + 1); v[i] = (4 + 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-3dcaebc50e901d48","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-7d4164eb4f384618"],"text":"A038231 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A038231 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}