{"id":"n-f930ec887abdd05e","revision":1,"first_index":0,"fingerprint":"f6-b7c219ffd1e0c541","name":"obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.","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":[["6","5"],["30","65","25"],["150","595","560","125"],["750","4745","8025","4335","625"],["3750","35095","93600","90320","31510","3125"],["18750","247445","966845","1463870","911620","219685","15625"],["93750","1687795","9211480","20326455","19829430","8547305","1486860","78125"],["468750","11236145","82829545","253845705","360025205","243227755","75970035","9843035","390625"],["2343750","73432495","713169320","2933559890","5759735200","5653555700","2773161680","648435430","64058210","1953125"],["11718750","473016845","5937042765","31952012160","83871878580","113791832490","81151882770","29892019800","5362039830","411302385","9765625"],["58593750","3011929195","48117867600","332158982705","1135572390260","2055090330590","2025170376680","1086372519650","308204005670","43228844615","2612345560","48828125"],["292968750","18999309545","381594661865","3325623635795","14508964938225","34099318769130","44774494606170","33209720782710","13754751401280","3065537497725","341377746445","16445557735","244140625"],["1464843750","118927497895","2972549005840","32286089518980","176836509711730","528552612717855","899764548740640","889089372903600","509851856873370","166407381946245","29602146683120","2650170568940","102774908910","1220703125"],["7324218750","739688034245","22813213775485","305510725078770","2072797212187900","7747785417374915","16736441399866815","21423419282281500","16375904639681070","7414770070213275","1938675919760635","278881127401490","20281970704440","638377969085","6103515625"],["36621093750","4576067658595","172904350276520","2829276660974075","23514083139434810","108406427692623775","292073451230678260","473797832814355335","469110854931940650","283445332008372105","103064454834598400","21880917944752945","2573208747348070","153362067606325","3945013908260","30517578125"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["11","120","1430","18480","257400","3843840","61261200","1037836800","18623404800","352864512000","7039647014400","147497366016000","3238237626624000","74338672472064000","1781030694643200000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["6","5"],["30","65","25"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","3"],["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","5"],["0","1"]]},"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":162837,"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-e045f1dbdfaaf32d"]}],"region_exact":{},"incoming_edges":[{"id":"e-e045f1dbdfaaf32d","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-f930ec887abdd05e","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":"a21ab69121442f69b0cc333bb40162b93cd85c149282256402b391ad55860ce0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-8919a5e7419ba068","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.","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-8919a5e7419ba068","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.","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:        6,         5;\n  1:       30,        65,         25;\n  2:      150,       595,        560,         125;\n  3:      750,      4745,       8025,        4335,         625;\n  4:     3750,     35095,      93600,       90320,       31510,         3125;\n  5:    18750,    247445,     966845,     1463870,      911620,       219685,       15625;\n  6:    93750,   1687795,    9211480,    20326455,    19829430,      8547305,     1486860,       78125;\n  7:   468750,  11236145,   82829545,   253845705,   360025205,    243227755,    75970035,     9843035,     390625;\n  8:  2343750,  73432495,  713169320,  2933559890,  5759735200,   5653555700,  2773161680,   648435430,   64058210,   1953125;\n  9: 11718750, 473016845, 5937042765, 31952012160, 83871878580, 113791832490, 81151882770, 29892019800, 5362039830, 411302385, 9765625;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+(n+3)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (4+(n+5)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 6 + 5*x, P(1,x) = 30 + 65*x + 25*x^2.","mathematica":"P[0, x_] := 6 + 5*x;\nP[1, x_] := 30 + 65*x + 25*x^2;\nP[n_, x_] := P[n, x] = Expand[(5 + (n + 3)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (4 + (n + 5)*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] = 6 + 5*x; v[2] = 30 + 65*x + 25*x^2; for(n = 2, N, my(i = n + 1); v[i] = (5 + (n + 3)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (4 + (n + 5)*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-8919a5e7419ba068","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-e045f1dbdfaaf32d"],"text":"A257607 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257607 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}