{"id":"n-32a9aa3e4733fa30","revision":1,"first_index":0,"fingerprint":"f6-2736c5ecf9748f6a","name":"obreschkoff_combination of Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 7.","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":[["2","1"],["2","17","1"],["2","159","144","1"],["2","1309","4433","1167","1"],["2","10523","94292","91168","9358","1"],["2","84249","1711385","4050090","1630740","74893","1"],["2","674071","28638240","138102131","135763426","27091749","599180","1"],["2","5392661","458034409","4057873841","7914380537","3950748655","431616731","4793483","1"],["2","43141395","7135954932","108779586090","374661499548","370804992852","105286871844","6709730862","38347914","1"],["2","345131281","109464829121","2746931517716","15515239566176","26782332056942","15253214159414","2648780600804","102798239654","306783369","1"],["2","2761050383","1663792403408","66625054087989","586554532324860","1628751086568102","1617483994077336","574048249683162","64085738316306","1561339295035","2454267016","1"],["2","22088403213","25150773144737","1571514584511683","20789214990792345","87856727224558818","139403200312182618","86825797762806342","20281791795197172","1509092638598785","23592194900085","19634136199","1"],["2","176707225867","378967312752564","36348098524951684","703016287292753762","4343827591341142947","10375957625201387976","10324411152113764176","4277791008569128950","684306010082068273","34864156659497636","355397206255692","157073089670","1"],["2","1413657807113","5699392366533769","829052489759205550","22957634605518818836","201229311013299978611","693160569848983324719","1034053832813022734772","687179986288210703490","197654536292023603423","22309676693961702247","794567300412460478","5344171867503944","1256584717445","1"],["2","11309262457095","85619842504633248","18721115482853056007","730193613792831727182","8869467664121303499267","42654777395959215099620","91141177024719511404579","90800283761755471994526","42168826720684304374741","8694770762434798714680","708712989588629230485","17932247574807129298","80277084294936417","10052677739652","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","20","306","6912","205344","7551360","330868800","16822840320","973460125440","63159412884480","4540779002073600","358286123999232000","30781256628084019200","2860180213579669094400","285805029214402465996800"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","1"],["2","17","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-13"],["0","7"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","7","-7"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"],["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":162863,"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-1b90717a181c7212"]}],"region_exact":{},"incoming_edges":[{"id":"e-1b90717a181c7212","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-32a9aa3e4733fa30","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":"1072f337976afbb18b8696b1342eeb1881331f1b191c9558f6b2bde527b66245","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-0af3ffe51a3d7181","role":"sequence","name":"Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 7.","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-0af3ffe51a3d7181","role":"sequence","name":"Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 7.","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: 2,         1;\n  1: 2,        17,            1;\n  2: 2,       159,          144,             1;\n  3: 2,      1309,         4433,          1167,              1;\n  4: 2,     10523,        94292,         91168,           9358,              1;\n  5: 2,     84249,      1711385,       4050090,        1630740,          74893,              1;\n  6: 2,    674071,     28638240,     138102131,      135763426,       27091749,         599180,             1;\n  7: 2,   5392661,    458034409,    4057873841,     7914380537,     3950748655,      431616731,       4793483,            1;\n  8: 2,  43141395,   7135954932,  108779586090,   374661499548,   370804992852,   105286871844,    6709730862,     38347914,         1;\n  9: 2, 345131281, 109464829121, 2746931517716, 15515239566176, 26782332056942, 15253214159414, 2648780600804, 102798239654, 306783369, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(7*n-13)*x)*P(n-2,x) + (7*x-7*x^2)*d/dx P(n-2,x) + (7*n+1)*x*P(n-1,x) + (7*x-7*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 2 + 17*x + x^2.","mathematica":"P[0, x_] := 2 + x;\nP[1, x_] := 2 + 17*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (7*n - 13)*x)*P[n - 2, x] + (7*x - 7*x^2)*D[P[n - 2, x], x] + (7*n + 1)*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] = 2 + x; v[2] = 2 + 17*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (7*n - 13)*x)*v[i - 2] + (7*x - 7*x^2)*deriv(v[i - 2], x) + (7*n + 1)*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-0af3ffe51a3d7181","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-1b90717a181c7212"],"text":"A142462 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A142462 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}