{"id":"n-b0ef7fa5c7dde9fc","revision":1,"first_index":0,"fingerprint":"f6-9a68980a3af14e80","name":"obreschkoff_combination of Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).","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","9","1"],["2","47","40","1"],["2","205","585","167","1"],["2","843","6028","5648","678","1"],["2","3401","52657","115122","47852","2725","1"],["2","13639","420984","1819315","1756850","376533","10916","1"],["2","54597","3197857","24787369","46161225","23451503","2834963","43683","1"],["2","218435","23553388","307511546","991564804","970190948","287257900","20773022","174754","1"],["2","873793","170203545","3584817852","18663828744","31254737382","18001149342","3321113340","149648958","699041","1"],["2","3495231","1215366760","40041079125","320642464860","851847204486","839256295848","306226133370","36890169330","1066591595","2796192","1"],["2","13980989","8613822361","433992637915","5159477869305","20630539352226","32040043633962","20111925521238","4893292815060","398317835345","7550725997","11184799","1"],["2","55924027","60763721580","4603757194676","79114784077210","458166799120131","1060168076053176","1048758571771440","443214351468198","74655483900065","4214053181692","53226976572","44739230","1"],["2","223696185","427381685665","48080894691062","1170016910210316","9531313083724171","31537936423698015","46302730739980740","30947344651046706","9167213516333183","1099965183504927","43927605230950","374210633120","178956957","1"],["2","894784823","3000485429368","496462421904903","16832350736474558","188548758860234707","864782484996494284","1804021638156027939","1789256843839745214","843366453429171749","180551562254111840","15780466447660693","452962198359618","2626498492433","715827868","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","12","90","960","13200","221760","4398240","100531200","2601244800","75147072000","2397191596800","83683779379200","3173009968128000","129849331003392000","5704095611934720000"],"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","9","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-5"],["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"],["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162816,"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-3646695af547e3ef"]}],"region_exact":{},"incoming_edges":[{"id":"e-3646695af547e3ef","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-b0ef7fa5c7dde9fc","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":"9c09e011dd7fd298c915460a85730ef5eea2ae0dc66f5ed7eac8c056e6314390","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-ddb0cd3a4d6edf09","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).","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-ddb0cd3a4d6edf09","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).","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,      9,         1;\n  2: 2,     47,        40,          1;\n  3: 2,    205,       585,        167,           1;\n  4: 2,    843,      6028,       5648,         678,           1;\n  5: 2,   3401,     52657,     115122,       47852,        2725,           1;\n  6: 2,  13639,    420984,    1819315,     1756850,      376533,       10916,          1;\n  7: 2,  54597,   3197857,   24787369,    46161225,    23451503,     2834963,      43683,         1;\n  8: 2, 218435,  23553388,  307511546,   991564804,   970190948,   287257900,   20773022,    174754,      1;\n  9: 2, 873793, 170203545, 3584817852, 18663828744, 31254737382, 18001149342, 3321113340, 149648958, 699041, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(3*n-5)*x)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (3*n+1)*x*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 2 + 9*x + x^2.","mathematica":"P[0, x_] := 2 + x;\nP[1, x_] := 2 + 9*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (3*n - 5)*x)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (3*n + 1)*x*P[n - 1, x] + (3*x - 3*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 + 9*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (3*n - 5)*x)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (3*n + 1)*x*v[i - 1] + (3*x - 3*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-ddb0cd3a4d6edf09","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-3646695af547e3ef"],"text":"A142458 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A142458 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}