{"id":"n-c0a8c0f6c6032261","revision":1,"first_index":0,"fingerprint":"f6-11be8f48840f0d47","name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*7^(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":[["8","1"],["56","15","1"],["392","161","22","1"],["2744","1519","315","29","1"],["19208","13377","3724","518","36","1"],["134456","112847","39445","7350","770","43","1"],["941192","924385","388962","90895","12740","1071","50","1"],["6588344","7411887","3647119","1025227","180075","20237","1421","57","1"],["46118408","58471553","32941720","10823708","2285752","321734","30184","1820","64","1"],["322828856","455419279","289063593","108707676","26823972","4537890","533022","42924","2268","71","1"],["2259801992","3510763809","2478864430","1050017325","296475480","58589202","8269044","833490","58800","2765","78","1"],["15818613944","26835148655","20862814819","9828985705","3125345685","706599894","116472510","14103474","1245090","78155","3311","85","1"],["110730297608","203664654529","172874852388","89665714754","31706405500","8071544943","1521907464","215196828","22819104","1792175","101332","3906","92","1"],["775112083256","1536382879311","1413788621245","800534855666","311610553254","88207220101","18724897191","3028285260","374930556","35364329","2501499","128674","4550","99","1"],["5425784582792","11529792238433","11432903228026","7017532610907","2981808728444","929061093961","219281500438","39922894011","5652799152","622480859","52874822","3402217","160524","5243","106","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["9","72","576","4608","36864","294912","2359296","18874368","150994944","1207959552","9663676416","77309411328","618475290624","4947802324992","39582418599936"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["8","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["7","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162791,"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-38dd62b72ba17f45"]}],"region_exact":{},"incoming_edges":[{"id":"e-38dd62b72ba17f45","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-c0a8c0f6c6032261","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":"51a715178c8d757f54c3dac9caa204075556b0cf01e9afdb2074690531d4f674","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-f211cb6153c37c6d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(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-f211cb6153c37c6d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(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-2f3158681bb76705","kind":"oeis_match","node_a_id":"n-c0a8c0f6c6032261","node_b_id":null,"fingerprint":null,"oeis_id":"A168686","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":"A168686","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-4c4f8e92ecd5297c","kind":"oeis_match","node_a_id":"n-c0a8c0f6c6032261","node_b_id":null,"fingerprint":null,"oeis_id":"A168926","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":"A168926","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-5d315b2de9ecb73d","kind":"oeis_match","node_a_id":"n-c0a8c0f6c6032261","node_b_id":null,"fingerprint":null,"oeis_id":"A167900","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":"A167900","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-638a1ce3fb33efb4","kind":"oeis_match","node_a_id":"n-c0a8c0f6c6032261","node_b_id":null,"fingerprint":null,"oeis_id":"A168878","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":"A168878","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-99e15d102307c3c8","kind":"oeis_match","node_a_id":"n-c0a8c0f6c6032261","node_b_id":null,"fingerprint":null,"oeis_id":"A168782","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":"A168782","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a60441fe98a0dd7b","kind":"oeis_match","node_a_id":"n-c0a8c0f6c6032261","node_b_id":null,"fingerprint":null,"oeis_id":"A168734","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":"A168734","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-bbe38323484ac13e","kind":"oeis_match","node_a_id":"n-c0a8c0f6c6032261","node_b_id":null,"fingerprint":null,"oeis_id":"A003951","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":"A003951","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-da9c12be977bec90","kind":"oeis_match","node_a_id":"n-c0a8c0f6c6032261","node_b_id":null,"fingerprint":null,"oeis_id":"A168830","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":"A168830","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A003951","relation":"match_row_sums"},{"oeis_id":"A167900","relation":"match_row_sums"},{"oeis_id":"A168686","relation":"match_row_sums"},{"oeis_id":"A168734","relation":"match_row_sums"},{"oeis_id":"A168782","relation":"match_row_sums"},{"oeis_id":"A168830","relation":"match_row_sums"},{"oeis_id":"A168878","relation":"match_row_sums"},{"oeis_id":"A168926","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:         8,         1;\n  1:        56,        15,         1;\n  2:       392,       161,        22,         1;\n  3:      2744,      1519,       315,        29,        1;\n  4:     19208,     13377,      3724,       518,       36,       1;\n  5:    134456,    112847,     39445,      7350,      770,      43,      1;\n  6:    941192,    924385,    388962,     90895,    12740,    1071,     50,     1;\n  7:   6588344,   7411887,   3647119,   1025227,   180075,   20237,   1421,    57,    1;\n  8:  46118408,  58471553,  32941720,  10823708,  2285752,  321734,  30184,  1820,   64,  1;\n  9: 322828856, 455419279, 289063593, 108707676, 26823972, 4537890, 533022, 42924, 2268, 71, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+x)*P(n-1,x) for n >= 1, with P(0,x) = 8 + x.","mathematica":"P[0, x_] := 8 + x;\nP[n_, x_] := P[n, x] = Expand[(7 + 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] = 8 + x; for(n = 1, N, my(i = n + 1); v[i] = (7 + 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-f211cb6153c37c6d","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-38dd62b72ba17f45"],"text":"A027466 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A027466 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}
