{"id":"n-540fd34dbc661863","revision":0,"first_index":0,"fingerprint":"f6-e1e7aa9d2da54c7d","name":"one_plus_derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["2160","5632","1936"],["60800","285120","371712","85184"],["1568000","10700800","25090560","21807104","3748096"],["38400000","344960000","1177088000","1839974400","1199390720","164916224"],["908800000","10137600000","45534720000","103583744000","121438310400","63327830016","7256313856"],["20992000000","279910400000","1561190400000","4674897920000","7975948288000","7480599920640","3250828607488","319277809664"],["476160000000","7389184000000","49264230400000","183179673600000","411391016960000","561506759475200","438861862010880","163470238547968","14048223625216"],["10649600000000","188559360000000","1463058432000000","6502878412800000","18134787686400000","32582168543232000","37059446125363200","24827042479472640","8091776808124416","618121839509504"],["235520000000000","4685824000000000","41483059200000000","214581903360000000","715316625408000000","1595861316403200000","2389359026503680000","2329450899308544000","1365487336370995200","395597977286082560","27197360938418176"],["5160960000000000","113991680000000000","1133969408000000000","6692600217600000000","25964410306560000000","69242649339494400000","128732812856524800000","165207109832540160000","140931779408166912000","73432874533729075200","19146942100646395904","1196683881290399744"],["112230400000000000","2724986880000000000","30093803520000000000","199578615808000000000","883423228723200000000","2741841728372736000000","6093353141875507200000","9710132169749299200000","10903669248947650560000","8267997725279125504000","3877255775380895170560","919053220831027003392","52654090776777588736"],["2424832000000000000","64195788800000000000","779346247680000000000","5737885204480000000000","28539742060544000000000","101063617365934080000000","261388911438200832000000","497913999593255731200000","694274450137074892800000","692988756710895124480000","472929469885965978828800","201617300319806548869120","43808203526278953828352","2316779994178213904384"],["52101120000000000000","1493696512000000000000","19772302950400000000000","160025762856960000000000","883634321489920000000000","3516096221859020800000000","10375864716235898880000000","23002224206561673216000000","38339377968680691302400000","47519229031604237107200000","42688107413391139667968000","26484050313614094814412800","10349688083083402841948160","2075834874783679658328064","101938319743841411792896"],["1114112000000000000000","34386739200000000000000","492919848960000000000000","4349906649088000000000000","26404250871398400000000000","116639730436669440000000000","386770584404492288000000000","978295816102241894400000000","1897683497041338040320000000","2811554384369917362176000000","3136269116085879649075200000","2561286444803468380078080000","1456622767248775214792704000","525445702679618913514291200","97860786954087755321180160","4485286068729022118887424"],["23724032000000000000000","784334848000000000000000","12104132198400000000000000","115671857889280000000000000","765583570239488000000000000","3717718522692894720000000000","13685728371235880960000000000","38898070202966081536000000000","86090031816997286707200000000","148441020213011331153920000000","197933428659642182297190400000","200721223429496297540812800000","150262138095136811631247360000","78881725241779827016466432000","26422412477603693936718643200","4592932934378518649740722176","197352587024076973231046656"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["9728","802816","62914560","4764729344","352187318272","25563645345792","1829587348619264","129478489286901760","9079256848778919936","631800984524552142848","43681889966544218226688","3003425083105094355910656","205517389334486959700049920","14004196694415864359796342784","950737950171172051122527404032"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2160","5632","1936"]],"recurrence":{"denominator":{"coeffs":[["16","11"],["11","0"]]},"terms":[{"coeff":{"coeffs":[["540","1408","484"],["220","484","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195152,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-1be3de9db1110d0e"]},{"id":195153,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-1be3de9db1110d0e"]},{"id":195154,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-1be3de9db1110d0e"]},{"id":195155,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-1be3de9db1110d0e"]}],"region_exact":{},"incoming_edges":[{"id":"e-1be3de9db1110d0e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-540fd34dbc661863","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"361fc31f906bec66fa7884ec4555d635b8ab2f18e10fb184eb7b3aea61c60393","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-586959989bac3453","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"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:             2160,               5632,                1936;\n  1:            60800,             285120,              371712,               85184;\n  2:          1568000,           10700800,            25090560,            21807104,              3748096;\n  3:         38400000,          344960000,          1177088000,          1839974400,           1199390720,            164916224;\n  4:        908800000,        10137600000,         45534720000,        103583744000,         121438310400,          63327830016,            7256313856;\n  5:      20992000000,       279910400000,       1561190400000,       4674897920000,        7975948288000,        7480599920640,         3250828607488,          319277809664;\n  6:     476160000000,      7389184000000,      49264230400000,     183179673600000,      411391016960000,      561506759475200,       438861862010880,       163470238547968,        14048223625216;\n  7:   10649600000000,    188559360000000,    1463058432000000,    6502878412800000,    18134787686400000,    32582168543232000,     37059446125363200,     24827042479472640,      8091776808124416,      618121839509504;\n  8:  235520000000000,   4685824000000000,   41483059200000000,  214581903360000000,   715316625408000000,  1595861316403200000,   2389359026503680000,   2329450899308544000,   1365487336370995200,   395597977286082560,    27197360938418176;\n  9: 5160960000000000, 113991680000000000, 1133969408000000000, 6692600217600000000, 25964410306560000000, 69242649339494400000, 128732812856524800000, 165207109832540160000, 140931779408166912000, 73432874533729075200, 19146942100646395904, 1196683881290399744;\n","recurrence_sentence":"The row polynomials satisfy (11*n+16+11*x)*P(n,x) = (220*n+540+(484*n+1408)*x+484*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2160 + 5632*x + 1936*x^2.","mathematica":"P[0, x_] := 2160 + 5632*x + 1936*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((220*n + 540 + (484*n + 1408)*x + 484*x^2)*P[n - 1, x])/(11*n + 16 + 11*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2160 + 5632*x + 1936*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((220*n + 540 + (484*n + 1408)*x + 484*x^2)*v[i - 1])/(11*n + 16 + 11*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-cfc5b48d32898a3e","steps":["basis A013611","D+1"],"edges":["e-9ac6ce8d07a161c9","e-1be3de9db1110d0e"],"text":"A038301 \u2192 basis A013611 \u2192 D+1"},"headline":"A038301 \u2192 basis A013611 \u2192 D+1","chain":null}