{"id":"n-469b72de5a90efc7","revision":1,"first_index":0,"fingerprint":"f6-e14527591166e5fd","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) = 4*n + 3.","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":[["4","3"],["12","45","9"],["36","435","402","27"],["108","3549","9039","3183","81"],["324","26787","151092","145602","23712","243"],["972","194637","2158341","4415826","2039526","171249","729"],["2916","1387731","28123878","106676085","104418600","26278989","1217454","2187"],["8748","9801597","346147371","2238926889","3993705495","2162632167","321256113","8587059","6561"],["26244","68908611","4106603688","42825450420","126966112032","125211333426","41008965624","3795149172","60330300","19683"],["78732","483357549","47550102465","768406313076","3560097193212","5784516034374","3477711310206","731503951140","43823861904","423053493","59049"],["236196","3386809587","541660776570","13175202025215","91299686671440","228674193770358","226449774929196","88590381674310","12491248176540","498320272215","2963834826","177147"],["708588","23718532125","6102209219271","218577746511795","2192915248092525","8076048653996994","12265701389987430","7940898565739238","2117500068443430","206628159111825","5607275756643","20754933495","531441"],["2125764","166065154275","68227217318748","3539069605151958","50140131889164480","262149702068019357","580639394793386520","576449742057396084","256401718750372644","48241209567232845","3338543655259404","62643660094230","145310929368","1594323"],["6377292","1162570871181","758885694348861","56270306043337062","1104017977868165634","7976488080906639711","24821091088196306463","35821592487824491908","24506791048206929736","7770226494045431547","1059315837906084747","52999989605396406","696408099408906","1017262067577","4782969"],["19131876","8138365981203","8411102794160718","882507269440496313","23607714206158874808","230739858591962397915","980235307871992952874","1976933168384806198089","1965932666985178012644","963705004904692662849","224073650075785132002","22604668674296127123","830266165529448768","7715857265622993","7121110290918","14348907"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["7","66","900","15960","347760","8981280","268107840","9081072000","344042899200","14414515315200","661726819353600","33030023791564800","1781030694643200000","103170250784240640000","6389560955349411840000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4","3"],["12","45","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","-5"],["0","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","4","-4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","3"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162854,"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-11ecc0603e43ec86"]}],"region_exact":{},"incoming_edges":[{"id":"e-11ecc0603e43ec86","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-469b72de5a90efc7","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":"692c077e9cb0a5255ee942e8981a7409c2297c76976cde185df82dc010962008","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-930be05ae3b321d9","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) = 4*n + 3.","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-930be05ae3b321d9","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) = 4*n + 3.","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:     4,         3;\n  1:    12,        45,           9;\n  2:    36,       435,         402,           27;\n  3:   108,      3549,        9039,         3183,            81;\n  4:   324,     26787,      151092,       145602,         23712,           243;\n  5:   972,    194637,     2158341,      4415826,       2039526,        171249,           729;\n  6:  2916,   1387731,    28123878,    106676085,     104418600,      26278989,       1217454,         2187;\n  7:  8748,   9801597,   346147371,   2238926889,    3993705495,    2162632167,     321256113,      8587059,        6561;\n  8: 26244,  68908611,  4106603688,  42825450420,  126966112032,  125211333426,   41008965624,   3795149172,    60330300,     19683;\n  9: 78732, 483357549, 47550102465, 768406313076, 3560097193212, 5784516034374, 3477711310206, 731503951140, 43823861904, 423053493, 59049;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(4*n-5)*x)*P(n-2,x) + (4*x-4*x^2)*d/dx P(n-2,x) + (2+(4*n+3)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 4 + 3*x, P(1,x) = 12 + 45*x + 9*x^2.","mathematica":"P[0, x_] := 4 + 3*x;\nP[1, x_] := 12 + 45*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(3 + (4*n - 5)*x)*P[n - 2, x] + (4*x - 4*x^2)*D[P[n - 2, x], x] + (2 + (4*n + 3)*x)*P[n - 1, x] + (4*x - 4*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] = 4 + 3*x; v[2] = 12 + 45*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (3 + (4*n - 5)*x)*v[i - 2] + (4*x - 4*x^2)*deriv(v[i - 2], x) + (2 + (4*n + 3)*x)*v[i - 1] + (4*x - 4*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-930be05ae3b321d9","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-11ecc0603e43ec86"],"text":"A257621 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257621 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}