{"id":"n-380c07af9e1e849e","revision":1,"first_index":0,"fingerprint":"f6-ad84a92684b7f675","name":"obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 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","38","4"],["12","424","392","8"],["24","4064","12804","3684","16"],["48","37240","295620","286880","33580","32"],["96","336824","5821388","13310588","5571748","303292","64"],["192","3035416","105369748","478802736","471313536","100134824","2732220","128"],["384","27328088","1817351452","14872376316","28412296472","14506432528","1720252044","24596060","256"],["768","245974168","30452230628","421329298208","1399386390388","1385846470232","408640217132","28759306672","221378492","512"],["1536","2213815640","501330020268","11231614981596","60419234626044","102857990195100","59454717454020","10853426995092","472816687764","1992437916","1024"],["3072","19924447768","8163628693236","287203142777072","2382540943962008","6454591881679760","6412153126110704","2334317841489584","276851045802248","7693025821112","17932011388","2048"],["6144","179320265432","132038672269244","7132999981766300","88058237753939952","359916153200916096","565426374490499592","355886541018767400","86011123055227440","6864312564657288","124365570895900","161388257116","4096"],["12288","1613882902936","2126659533204484","173511946750324512","3103457482279559020","18408011006117976144","43190012553509225448","42984502332219641952","18140182702853148696","3024579359206097880","166779672857936468","2002473282574016","1452494651964","8192"],["24576","14524947240536","34164216743138764","4157910444908419260","105548588283691220404","882221000362303310588","2965550249374161600888","4390986746288058217752","2940973758048059935080","867184275141729720912","102697520871235447036","3993257431583687276","32163357343467684","13072452600860","16384"],["49152","130724527565080","547968120521054996","98556046689543344048","3493765986381968377136","40220551310667266590744","187702194860305086729460","395643821949401891398368","394218385452487994781504","185643442102619330104968","39458915623809228090236","3395226375955641349360","94596953474702795536","515819288124827848","117652074988796","32768"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["5","48","836","20592","653400","25344000","1161388800","61380633600","3674881267200","245795337216000","18163552492800000","1469552487017472000","129195169254987264000","12263379661417218048000","1249970520697577902080000"],"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","38","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","-12"],["0","7"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","7","-7"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"],["0","7"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","7","-7"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162879,"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-b3016eed90e91248"]}],"region_exact":{},"incoming_edges":[{"id":"e-b3016eed90e91248","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-380c07af9e1e849e","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":"0b6ab03ae3ca762907fdaf63c7d73a4f5913e2a507fc7374eb6776a5e224ae25","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-5b7da9f56b3380cb","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 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-5b7da9f56b3380cb","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 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,         38,            4;\n  2:   12,        424,          392,              8;\n  3:   24,       4064,        12804,           3684,             16;\n  4:   48,      37240,       295620,         286880,          33580,              32;\n  5:   96,     336824,      5821388,       13310588,        5571748,          303292,             64;\n  6:  192,    3035416,    105369748,      478802736,      471313536,       100134824,        2732220,            128;\n  7:  384,   27328088,   1817351452,    14872376316,    28412296472,     14506432528,     1720252044,       24596060,          256;\n  8:  768,  245974168,  30452230628,   421329298208,  1399386390388,   1385846470232,   408640217132,    28759306672,    221378492,        512;\n  9: 1536, 2213815640, 501330020268, 11231614981596, 60419234626044, 102857990195100, 59454717454020, 10853426995092, 472816687764, 1992437916, 1024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(7*n-12)*x)*P(n-2,x) + (7*x-7*x^2)*d/dx P(n-2,x) + (1+(7*n+2)*x)*P(n-1,x) + (7*x-7*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 6 + 38*x + 4*x^2.","mathematica":"P[0, x_] := 3 + 2*x;\nP[1, x_] := 6 + 38*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + (7*n - 12)*x)*P[n - 2, x] + (7*x - 7*x^2)*D[P[n - 2, x], x] + (1 + (7*n + 2)*x)*P[n - 1, x] + (7*x - 7*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 + 38*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (2 + (7*n - 12)*x)*v[i - 2] + (7*x - 7*x^2)*deriv(v[i - 2], x) + (1 + (7*n + 2)*x)*v[i - 1] + (7*x - 7*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-5b7da9f56b3380cb","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-b3016eed90e91248"],"text":"A257617 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257617 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}