{"id":"n-13bed6262cd8149a","revision":1,"first_index":0,"fingerprint":"f6-972b37de3d251217","name":"obreschkoff_combination of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*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","33","9"],["36","243","222","27"],["108","1521","3195","1335","81"],["324","8739","35628","33954","7512","243"],["972","47745","342909","620274","317694","40557","729"],["2916","252819","3006186","9289173","9011808","2736237","213234","2187"],["8748","1312209","24754959","122062449","200135535","116056827","22271541","1101891","6561"],["26244","6722883","195167256","1464494580","3770899632","3692826546","1373192712","174207924","5629740","19683"],["78732","34152417","1491706161","16454176500","63220592844","96645572334","60992260878","15276151044","1324221120","28548921","59049"],["236196","172533555","11147897526","176083789119","972445254432","2197542401718","2164117422228","927907170054","162299344236","9857828823","144063366","177147"],["708588","868454577","81946495971","1816222754235","14012283706365","44980338293274","65238765145806","43788666125478","13260460325430","1665071307525","72261200703","724627407","531441"],["2125764","4361050467","595048827588","18211115596278","191953837116360","849152179979037","1739038275582696","1718779848730164","819375797559924","180509599545165","16627990784724","523656081462","3637131648","1594323"],["6377292","21865836609","4281641255205","178632026663430","2527037382978474","15043809960012771","42133713829345647","58656639535617060","41264086160594520","14416387999569207","2364738899029587","162599084706294","3762346224210","18230830725","4782969"],["19131876","109523690451","30598324847682","1722246690599001","32230100132172288","253333680573596475","946313163964491318","1794351626889593481","1777828701183322068","920092887763909569","241444329957005742","30043993259806131","1563929247147912","26857844460081","91299237018","14348907"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["7","54","528","6240","86400","1370880","24514560","487710720","10683187200","255467520000","6621718118400","184917609676800","5534775351705600","176755728973824000","5998982316687360000"],"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","33","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","-1"],["0","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","3"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162828,"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-cf5b10dddbff9e73"]}],"region_exact":{},"incoming_edges":[{"id":"e-cf5b10dddbff9e73","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-13bed6262cd8149a","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":"4c961efe50540619d51e33175409305edef27842ca68fff52d604d64b2a83e7d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-0c0e99d973615582","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*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-0c0e99d973615582","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*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,       33,          9;\n  2:    36,      243,        222,          27;\n  3:   108,     1521,       3195,        1335,          81;\n  4:   324,     8739,      35628,       33954,        7512,         243;\n  5:   972,    47745,     342909,      620274,      317694,       40557,         729;\n  6:  2916,   252819,    3006186,     9289173,     9011808,     2736237,      213234,        2187;\n  7:  8748,  1312209,   24754959,   122062449,   200135535,   116056827,    22271541,     1101891,       6561;\n  8: 26244,  6722883,  195167256,  1464494580,  3770899632,  3692826546,  1373192712,   174207924,    5629740,    19683;\n  9: 78732, 34152417, 1491706161, 16454176500, 63220592844, 96645572334, 60992260878, 15276151044, 1324221120, 28548921, 59049;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(2*n-1)*x)*P(n-2,x) + (2*x-2*x^2)*d/dx P(n-2,x) + (2+(2*n+3)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 4 + 3*x, P(1,x) = 12 + 33*x + 9*x^2.","mathematica":"P[0, x_] := 4 + 3*x;\nP[1, x_] := 12 + 33*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(3 + (2*n - 1)*x)*P[n - 2, x] + (2*x - 2*x^2)*D[P[n - 2, x], x] + (2 + (2*n + 3)*x)*P[n - 1, x] + (2*x - 2*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 + 33*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (3 + (2*n - 1)*x)*v[i - 2] + (2*x - 2*x^2)*deriv(v[i - 2], x) + (2 + (2*n + 3)*x)*v[i - 1] + (2*x - 2*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-0c0e99d973615582","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-cf5b10dddbff9e73"],"text":"A257611 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257611 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}