{"id":"n-3fbde8aced8e33e7","revision":1,"first_index":0,"fingerprint":"f6-818396bdab1dff2f","name":"obreschkoff_combination of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*n + 2.","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","2"],["6","30","4"],["12","272","248","8"],["24","2088","6100","1852","16"],["48","15104","107476","103464","13276","32"],["96","106944","1612780","3556700","1524420","93716","64"],["192","751520","22174548","95436960","93493040","20757072","657900","128"],["384","5267424","289674796","2213585316","4139184600","2142266824","269669244","4609716","256"],["768","36887456","3667703060","46797226904","150368955332","148443356048","44944634876","3403301240","32278124","512"],["1536","258247008","45538661932","929912813548","4795020261052","8047696393660","4693374131908","888635881732","42172248148","225969652","1024"],["3072","1807806368","558440163220","17704148173200","139673841727080","369996842219424","366744166086816","135910380408720","16861050953400","516526162080","1581838252","2048"],["6144","12654814560","6794158044076","327006506764340","3811972555319760","15134603391284208","23536205392594440","14907533066924472","3693766323084240","310703895974160","6279439688364","11072979380","4096"],["12288","88584072608","82243311772692","5909825332321032","99148607480043100","568062013172010336","1311328420702645896","1302926818701335664","556966452660405624","95781847663723200","5605915594390948","75976526898888","77511099372","8192"],["24576","620089311072","992356588887340","105123510131245308","2487029343862735764","19979274170145799172","65747251476090573624","96611573308322237400","65012386670561248296","19521456324965436648","2397157321531333596","99599837848627052","916468646061060","542578223988","16384"],["49152","4340626906016","11949437495013524","1848245953472577536","60678687453778470304","668288570174459127472","3040171485705509394532","6329572002849838726848","6298707681017607063072","2995138373205093945408","651254143193356426588","58386653642378382016","1749638472212837824","11033588938263536","3798048706604","32768"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["5","40","540","10080","239400","6894720","233271360","9064258560","397694344320","19442834611200","1047968785543680","61734888457482240","3945888287240739840","271962761951361761280","20105818472832815923200"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3","2"],["6","30","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","-8"],["0","5"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","5","-5"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"],["0","5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","5","-5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162860,"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-18b3eb6c762aa18f"]}],"region_exact":{},"incoming_edges":[{"id":"e-18b3eb6c762aa18f","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-3fbde8aced8e33e7","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":"6ca6c100c5290b654d85094b23c89e793750ee4d858db72d5c9dcf7ca4453a24","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-ee6acb764200c6c7","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*n + 2.","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-ee6acb764200c6c7","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*n + 2.","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,         2;\n  1:    6,        30,           4;\n  2:   12,       272,         248,            8;\n  3:   24,      2088,        6100,         1852,            16;\n  4:   48,     15104,      107476,       103464,         13276,            32;\n  5:   96,    106944,     1612780,      3556700,       1524420,         93716,            64;\n  6:  192,    751520,    22174548,     95436960,      93493040,      20757072,        657900,          128;\n  7:  384,   5267424,   289674796,   2213585316,    4139184600,    2142266824,     269669244,      4609716,         256;\n  8:  768,  36887456,  3667703060,  46797226904,  150368955332,  148443356048,   44944634876,   3403301240,    32278124,       512;\n  9: 1536, 258247008, 45538661932, 929912813548, 4795020261052, 8047696393660, 4693374131908, 888635881732, 42172248148, 225969652, 1024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(5*n-8)*x)*P(n-2,x) + (5*x-5*x^2)*d/dx P(n-2,x) + (1+(5*n+2)*x)*P(n-1,x) + (5*x-5*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 6 + 30*x + 4*x^2.","mathematica":"P[0, x_] := 3 + 2*x;\nP[1, x_] := 6 + 30*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + (5*n - 8)*x)*P[n - 2, x] + (5*x - 5*x^2)*D[P[n - 2, x], x] + (1 + (5*n + 2)*x)*P[n - 1, x] + (5*x - 5*x^2)*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 + 2*x; v[2] = 6 + 30*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (2 + (5*n - 8)*x)*v[i - 2] + (5*x - 5*x^2)*deriv(v[i - 2], x) + (1 + (5*n + 2)*x)*v[i - 1] + (5*x - 5*x^2)*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-ee6acb764200c6c7","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-18b3eb6c762aa18f"],"text":"A257614 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257614 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}