{"id":"n-92ef49366ed0ce97","revision":1,"first_index":0,"fingerprint":"f6-ad1d7f86469919e6","name":"obreschkoff_combination of Triangle read by rows: A008277 * A007318.","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":false,"rows":[["3","1"],["7","6","1"],["20","27","10","1"],["67","121","70","15","1"],["255","570","446","149","21","1"],["1080","2857","2801","1275","280","28","1"],["5017","15253","17886","10296","3095","482","36","1"],["25287","86548","117685","81746","31276","6685","777","45","1"],["137122","520301","802526","651519","302827","82733","13216","1190","55","1"],["794545","3303077","5684962","5271429","2882146","961254","196791","24366","1749","66","1"],["4892167","22073700","41857212","43570691","27370575","10791670","2706144","430077","42450","2485","78","1"],["31858034","154827691","320214621","369192967","261752491","119149129","35451361","6916437","876942","70565","3432","91","1"],["218543759","1136770349","2543265076","3213189420","2535453558","1308504560","451175076","104714830","16320942","1687477","112750","4627","105","1"],["1573857867","8715384958","20949398913","28751026408","24968433568","14402035934","5650908978","1524176082","283111543","36010689","3091374","174161","6110","120","1"],["11863100692","69618810567","178766059834","264578265225","250571807486","159688138974","70278540332","21633386060","4679387713","710142862","75022948","5428787","261261","7924","136","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["4","14","58","274","1442","8322","52066","350050","2511490","19120386","153737250","1300313762","11529742530","106857616706","1032468360802"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162739,"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-e25ba294e22c6482"]}],"region_exact":{},"incoming_edges":[{"id":"e-e25ba294e22c6482","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-92ef49366ed0ce97","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":"d3ab8942a4cc0d8c53c9bdfd3ea50ca1d3081dd51561f273087417d29e1db2f2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-bdae5aa3d1cac0ec","role":"sequence","name":"Triangle read by rows: A008277 * A007318.","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-bdae5aa3d1cac0ec","role":"sequence","name":"Triangle read by rows: A008277 * A007318.","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":[],"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:      3,       1;\n  1:      7,       6,       1;\n  2:     20,      27,      10,       1;\n  3:     67,     121,      70,      15,       1;\n  4:    255,     570,     446,     149,      21,      1;\n  5:   1080,    2857,    2801,    1275,     280,     28,      1;\n  6:   5017,   15253,   17886,   10296,    3095,    482,     36,     1;\n  7:  25287,   86548,  117685,   81746,   31276,   6685,    777,    45,    1;\n  8: 137122,  520301,  802526,  651519,  302827,  82733,  13216,  1190,   55,  1;\n  9: 794545, 3303077, 5684962, 5271429, 2882146, 961254, 196791, 24366, 1749, 66, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3 + x.","mathematica":"P[0, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3 + x; for(n = 1, N, my(i = n + 1); v[i] = (2 + x)*v[i - 1] + (1 + x)*deriv(v[i - 1], x)); 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-c8c19eba9d68881f","steps":["basis A049020","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-3c112fda27f20cb0","e-e25ba294e22c6482"],"text":"A007318 \u2192 basis A049020 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A007318 \u2192 basis A049020 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}