{"id":"n-f055c1e733379fba","revision":1,"first_index":0,"fingerprint":"f6-f985e5e5089e1527","name":"obreschkoff_combination of Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.","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","3"],["2","15","15"],["2","45","156","105"],["2","111","927","1935","945"],["2","249","4236","19368","27630","10395"],["2","531","16707","142890","423180","446355","135135"],["2","1101","60228","878643","4635306","9792135","8056440","2027025"],["2","2247","204927","4799817","41251977","150269445","241137765","160826715","34459425"],["2","4545","670980","24181530","319195620","1837695708","4965823620","6326742870","3520588050","654729075"],["2","9147","2140227","115039692","2236739832","19214220870","80280418230","169094060460","176723979390","83896849575","13749310575"],["2","18357","6704268","524966733","14574484908","179304232986","1094870764656","3506125954410","5968457556930","5246609594625","2162802618900","316234143225"],["2","36783","20736975","2322930435","89911612665","1536345242838","13169974259862","60668487141750","154904992338420","219063193987995","165199561501275","59986429878375","7905853580625"],["2","73641","63571932","10043758620","531893372850","12327118117713","143940197691864","916899078848688","3325506920070390","6975301043252475","8373600893731500","5504177282152500","1781447430712950","213458046676875"],["2","147363","193661475","42670932030","3045494845260","93942781819173","1459561078853577","12480587793777780","61849054178423730","182372916397021245","321652844778836565","333516204460659150","193609301758790400","56408118014122875","6190283353629375"],["2","294813","587321076","178867435479","16995200700870","687158174507553","13941781145733672","156372215786354979","1028325102394737006","4102614977129576055","10083322079209751220","15235412369630573925","13840138353267434250","7173499985506406475","1897273652616342000","191898783962510625"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["5","32","308","3920","61880","1164800","25450880","632952320","17649632000","545312768000","18489511040000","682527093248000","27247200012032000","1169635902955520000","53723503633479680000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","3"],["2","15","15"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-3"],["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","3"],["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162832,"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-b4f1d8e4113d9eff"]}],"region_exact":{},"incoming_edges":[{"id":"e-b4f1d8e4113d9eff","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-f055c1e733379fba","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":"d81dccdde7bed2eb81ac3adbae023126d7cd1a8a1bea25ace5283a460f7ffb18","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-86fb6463ff8368f2","role":"sequence","name":"Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.","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-86fb6463ff8368f2","role":"sequence","name":"Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.","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,    3;\n  1: 2,   15,      15;\n  2: 2,   45,     156,       105;\n  3: 2,  111,     927,      1935,        945;\n  4: 2,  249,    4236,     19368,      27630,       10395;\n  5: 2,  531,   16707,    142890,     423180,      446355,      135135;\n  6: 2, 1101,   60228,    878643,    4635306,     9792135,     8056440,      2027025;\n  7: 2, 2247,  204927,   4799817,   41251977,   150269445,   241137765,    160826715,     34459425;\n  8: 2, 4545,  670980,  24181530,  319195620,  1837695708,  4965823620,   6326742870,   3520588050,   654729075;\n  9: 2, 9147, 2140227, 115039692, 2236739832, 19214220870, 80280418230, 169094060460, 176723979390, 83896849575, 13749310575;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(3*n-3)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (3*n+3)*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 3*x, P(1,x) = 2 + 15*x + 15*x^2.","mathematica":"P[0, x_] := 2 + 3*x;\nP[1, x_] := 2 + 15*x + 15*x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (3*n - 3)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (3*n + 3)*x*P[n - 1, x] + (x - 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 + 3*x; v[2] = 2 + 15*x + 15*x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (3*n - 3)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (3*n + 3)*x*v[i - 1] + (x - 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-86fb6463ff8368f2","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-b4f1d8e4113d9eff"],"text":"A219512 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A219512 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}