{"id":"n-deb7177d421fbe93","revision":1,"first_index":1,"fingerprint":"f6-328f070cb3b06156","name":"obreschkoff_combination of Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["0","1"],["0","-3","2"],["0","5","-16","6"],["0","-7","54","-90","24"],["0","9","-128","504","-576","120"],["0","-11","250","-1800","4800","-4200","720"],["0","13","-432","4950","-24000","48600","-34560","5040"],["0","-15","686","-11466","88200","-323400","529200","-317520","40320"],["0","17","-1024","23520","-263424","1528800","-4515840","6209280","-3225600","362880"],["0","-19","1458","-44064","677376","-5715360","26671680","-66044160","78382080","-35925120","3628800"],["0","21","-2000","76950","-1555200","17992800","-121927680","476280000","-1016064000","1061424000","-435456000","39916800"],["0","-23","2662","-127050","3267000","-49658400","461039040","-2612554560","8781696000","-16465680000","15367968000","-5708102400","479001600"],["0","25","-3456","200376","-6388800","123492600","-1505433600","11679655680","-56905390080","167949936000","-281014272000","237105792000","-80472268800","6227020800"],["0","-27","4394","-304200","11778624","-282196200","4372814160","-44523198720","296821324800","-1268911163520","3339239904000","-5045962521600","3885660979200","-1214269056000","87178291200"],["0","29","-5488","447174","-20669376","601200600","-11543051520","149338229040","-1306013829120","7635728580480","-29088489830400","69085163347200","-95198693990400","67432408243200","-19527937228800","1307674368000"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["1","-1","-5","-19","-71","-241","-389","6005","118609","1632671","20685691","256851869","3182340745","39336656111","478215810619"],"nonn":false,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0","1"]],"recurrence":{"denominator":{"coeffs":[["-3","1"],["2","-2"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["1","2","-1"],["-2","-6","4"],["0","3","-5"],["0","0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","-1"],["0","0","-2","2"],["0","0","0","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162758,"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-1c58af07f6ac5587"]}],"region_exact":{},"incoming_edges":[{"id":"e-1c58af07f6ac5587","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-deb7177d421fbe93","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":"40d93eab461b9a8b0f404a6f305ecdb1f9f2ca18cefe5ec48c118b264838ec8b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-3de13f8c869a1c34","role":"sequence","name":"Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).","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-3de13f8c869a1c34","role":"sequence","name":"Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).","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: 0,   1;\n   2: 0,  -3,     2;\n   3: 0,   5,   -16,      6;\n   4: 0,  -7,    54,    -90,      24;\n   5: 0,   9,  -128,    504,    -576,      120;\n   6: 0, -11,   250,  -1800,    4800,    -4200,      720;\n   7: 0,  13,  -432,   4950,  -24000,    48600,   -34560,      5040;\n   8: 0, -15,   686, -11466,   88200,  -323400,   529200,   -317520,    40320;\n   9: 0,  17, -1024,  23520, -263424,  1528800, -4515840,   6209280, -3225600,    362880;\n  10: 0, -19,  1458, -44064,  677376, -5715360, 26671680, -66044160, 78382080, -35925120, 3628800;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3+(n^2-2*n+1)*x)*P(n,x) = -(2*n-1-(3*n^2-6*n+2)*x-(2*n^3-5*n^2+4*n-1)*x^2)*P(n-1,x) - ((2*n-1)*x^2+(n^2-2*n+1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = x.","mathematica":"P[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 - (3*n^2 - 6*n + 2)*x - (2*n^3 - 5*n^2 + 4*n - 1)*x^2)*P[n - 1, x] - ((2*n - 1)*x^2 + (n^2 - 2*n + 1)*x^3)*D[P[n - 1, x], x])/(2*n - 3 + (n^2 - 2*n + 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = (-(2*n - 1 - (3*n^2 - 6*n + 2)*x - (2*n^3 - 5*n^2 + 4*n - 1)*x^2)*v[i - 1] - ((2*n - 1)*x^2 + (n^2 - 2*n + 1)*x^3)*deriv(v[i - 1], x))/(2*n - 3 + (n^2 - 2*n + 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-47fe395e9f7d5f86","steps":["x \u21a6 \u2212x (shift -1)","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-e31dcd4b09a7ff12","e-1c58af07f6ac5587"],"text":"A144084 \u2192 x \u21a6 \u2212x (shift -1) \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A144084 \u2192 x \u21a6 \u2212x (shift -1) \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}