{"id":"n-576ec26b3dfabfc3","revision":1,"first_index":1,"fingerprint":"f6-58f7968dbeed8539","name":"obreschkoff_combination of Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537 (S2[2,1] generalized Stirling2), for n >= 0.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2","2"],["2","18","12"],["2","82","296","120"],["2","290","3156","6072","1680"],["2","930","23952","122288","147744","30240"],["2","2866","153308","1647408","4950720","4160544","665280"],["2","8690","892728","17802152","106227296","212921952","133223040","17297280"],["2","26178","4915620","168193640","1762696080","6765250656","9789891264","4783720320","518918400"],["2","78658","26148896","1456377440","24816552640","161741180096","436799783424","481870775808","190464606720","17643225600"],["2","236114","136093356","11887228256","312428994560","3205222846656","14359969237632","28990792140288","25365822617088","8330583421440","670442572800"],["2","708498","698411592","93105555928","3633531289920","55692522720576","384161938813184","1264322958287616","1992904925243904","1425052650355200","397144985610240","28158588057600"],["2","2125666","3551571956","708095845400","39879352784656","879184875657536","8868181490007168","44208069721682688","112039297640678400","142463079005465088","85226384295429120","20499533265254400","1295295050649600"],["2","6377186","17953421232","5271637897744","419150235505888","12914386499978208","183441639794620928","1316404643189566976","4989346466245320192","10085650313002089984","10611131050861572096","5411447757891440640","1139131903885516800","64764752532480000"],["2","19131762","90404824956","38637162485392","4262881497438144","179489491933609440","3488766217199697728","34720300491978567168","187154674011347206656","559958421900454497792","927700903263876867072","824119308510347759616","363824287588989665280","67807936383331737600","3497296636753920000"],["2","57395506","454090355288","279923159552776","42269755531774112","2389085876896275616","62129270312484032384","833393917428318189184","6157387841441152214528","25946859085857882748416","63078919044987210762240","87520400528136013191168","66740830974383370854400","25834114570352595886080","4304995540212171571200","202843204931727360000"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["4","32","500","11200","325156","11580128","488373140","23793612160","1314818729284","81247285388192","5551165905054260","415519616347151680","33814657353735787876","2972455686105297408608","280684305614052662162900"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","2"],["2","18","12"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-22"],["0","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2","-4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-6"],["0","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162874,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-8644a9f1d4c74332"]}],"region_exact":{},"incoming_edges":[{"id":"e-8644a9f1d4c74332","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-576ec26b3dfabfc3","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":"2f380ffc2a1b2636329b227b3bb18ded217558567177b03c56df6670927c0d4c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-945ed98594b7abe2","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537 (S2[2,1] generalized Stirling2), for n >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-945ed98594b7abe2","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537 (S2[2,1] generalized Stirling2), for n >= 0.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 2,      2;\n   2: 2,     18,        12;\n   3: 2,     82,       296,         120;\n   4: 2,    290,      3156,        6072,         1680;\n   5: 2,    930,     23952,      122288,       147744,         30240;\n   6: 2,   2866,    153308,     1647408,      4950720,       4160544,         665280;\n   7: 2,   8690,    892728,    17802152,    106227296,     212921952,      133223040,       17297280;\n   8: 2,  26178,   4915620,   168193640,   1762696080,    6765250656,     9789891264,     4783720320,      518918400;\n   9: 2,  78658,  26148896,  1456377440,  24816552640,  161741180096,   436799783424,   481870775808,   190464606720,   17643225600;\n  10: 2, 236114, 136093356, 11887228256, 312428994560, 3205222846656, 14359969237632, 28990792140288, 25365822617088, 8330583421440, 670442572800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(8*n-22)*x)*P(n-2,x) + (2*x-4*x^2)*d/dx P(n-2,x) + (8*n-6)*x*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + 2*x, P(2,x) = 2 + 18*x + 12*x^2.","mathematica":"P[1, x_] := 2 + 2*x;\nP[2, x_] := 2 + 18*x + 12*x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (8*n - 22)*x)*P[n - 2, x] + (2*x - 4*x^2)*D[P[n - 2, x], x] + (8*n - 6)*x*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2 + 2*x; v[2] = 2 + 18*x + 12*x^2; for(n = 3, N, my(i = n); v[i] = (1 + (8*n - 22)*x)*v[i - 2] + (2*x - 4*x^2)*deriv(v[i - 2], x) + (8*n - 6)*x*v[i - 1] + (2*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-945ed98594b7abe2","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-8644a9f1d4c74332"],"text":"A290315 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A290315 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}