{"id":"n-7e6f860aed157553","revision":1,"first_index":1,"fingerprint":"f6-08170c9a4897cc91","name":"obreschkoff_combination of Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.","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":[["3","1"],["6","14","1"],["12","106","61","1"],["24","638","1191","252","1"],["48","3478","15629","10929","1019","1"],["96","18110","168611","273116","90016","4090","1"],["192","92278","1629849","5152725","3959350","696588","16377","1"],["384","465422","14746543","82295220","121629011","51168502","5191389","65528","1"],["768","2336326","127980325","1181004529","3004858813","2439430621","613435303","37797739","262135","1"],["1536","11702366","1081085787","15766804548","64274894634","89194785588","43829876124","6988175988","271203102","1048566","1"],["3072","58557910","8970508529","200117866149","1243407426144","2728873052580","2296547228034","728691656466","76819843896","1928043726","4194293","1"],["6144","292890926","73550100359","2449571637644","22366661004885","73561641885588","97519115066790","53374998899760","11449456144026","823298884110","13627377763","16777204","1"],["12288","1464675814","598212683709","29201328728433","381164895938831","1806168378282129","3560812221153906","3073728184126890","1149520618502766","172488950802744","8663283944033","95966263605","67108851","1"],["24576","7323858302","4839162212659","341357209354572","6234757382795716","41313925592963798","116157695156139411","148747819731940632","88130411806421616","23360390747600538","2517117330977881","89950789116092","674263649964","268435442","1"],["49152","36620323702","39002590268041","3932380304621845","98811936841939514","894534273183354932","3474738073314920111","6314745593443021863","5554779516290336460","2348514712868641302","453799003816159415","35844853939493987","924842061400346","4730650076320","1073741809","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["4","21","180","2106","31104","554040","11547360","275562000","7407106560","221419578240","7285418380800","261632230675200","10182443572224000","426898946765491200","19180668956998348800"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["3","1"],["6","14","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","-8"],["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","-2"],["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":162825,"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-998f3ee2c5950862"]}],"region_exact":{},"incoming_edges":[{"id":"e-998f3ee2c5950862","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-7e6f860aed157553","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":"f1478dc3d6286be933d1fe81f63f9a4d71568614eb1e9f55791a3e1444f04489","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-daa3b3e1ead00edc","role":"sequence","name":"Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending 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-daa3b3e1ead00edc","role":"sequence","name":"Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending 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:    3,        1;\n   2:    6,       14,          1;\n   3:   12,      106,         61,           1;\n   4:   24,      638,       1191,         252,           1;\n   5:   48,     3478,      15629,       10929,        1019,           1;\n   6:   96,    18110,     168611,      273116,       90016,        4090,           1;\n   7:  192,    92278,    1629849,     5152725,     3959350,      696588,       16377,          1;\n   8:  384,   465422,   14746543,    82295220,   121629011,    51168502,     5191389,      65528,         1;\n   9:  768,  2336326,  127980325,  1181004529,  3004858813,  2439430621,   613435303,   37797739,    262135,       1;\n  10: 1536, 11702366, 1081085787, 15766804548, 64274894634, 89194785588, 43829876124, 6988175988, 271203102, 1048566, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(3*n-8)*x)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (1+(3*n-2)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 3 + x, P(2,x) = 6 + 14*x + x^2.","mathematica":"P[1, x_] := 3 + x;\nP[2, x_] := 6 + 14*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + (3*n - 8)*x)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (1 + (3*n - 2)*x)*P[n - 1, x] + (3*x - 3*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] = 3 + x; v[2] = 6 + 14*x + x^2; for(n = 3, N, my(i = n); v[i] = (2 + (3*n - 8)*x)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (1 + (3*n - 2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-daa3b3e1ead00edc","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-998f3ee2c5950862"],"text":"A225117 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A225117 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}