{"id":"n-599179c3856533fe","revision":0,"first_index":0,"fingerprint":"f6-e470dede5399547f","name":"one_plus_derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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":[["403","396","81"],["6760","10881","5346","729"],["107653","243360","195858","64152","6561"],["1656538","4844385","5475600","2937870","721710","59049"],["24876631","89453052","130798395","98560800","39661245","7794468","531441"],["366837484","1567227753","2817771138","2746766295","1552332600","499731687","81841914","4782969"],["5333623945","26412298848","56420199108","67626507312","49441793310","22353589440","5996780244","841802544","43046721"],["76678687774","432023539545","1069698103344","1523345375916","1369436773068","800957051622","301773457440","69391314252","8523250758","387420489"],["1092263435419","6901081899660","19441059279525","32090943100320","34275270958110","24649861915224","12014355774330","3879944452800","780652285335","85232507580","3486784401"],["15440151087088","108134080106481","341603554033170","641554956224325","794250841732920","678650364970578","406722721601196","169917317379810","48014312603400","8587175138685","843801825042","31381059609"],["216851407678477","1667536317405504","5839240325749974","12297727945194120","17321983818056775","17155818181431072","12215706569470404","6275150561847024","2293883784627435","576171751240800","92741491497798","8284599736776","282429536481"],["3028751065922530","25371614698381809","97550874568221984","227730372704248986","359708542396928010","405334421342528535","334538454537905904","204176809804005324","91774076967012726","29820489200156655","6741209489517360","986432227749306","80774847433566","2541865828329"],["42099639816323167","381622634306238780","1598411725998053967","4097136731865323328","7173506740183843059","9064655268402585852","8512022848193099235","6021692181682306272","3215784754413083853","1284837077538178164","375738163921973853","77217490516289760","10357538391367713","782894675125332","22876792454961"],["582731705083494772","5683451375203627545","25759527815671117650","71928527669912428515","138278364700454662320","193684681984963762593","203954743539058181670","164160440643724056675","101616055565888918340","48236771316196257795","17345300546765405214","4611332011769679105","868696768308259800","107559052525741635","7549341510137130","205891132094649"],["8036185203212248849","83913365532023247168","409208499014661183240","1236457335152213647200","2589426996116847426540","3982416903373094274816","4648432367639130302232","4195640438517768308640","2954887931587033020150","1625856889054222693440","694609506953226112248","227065752612201668256","55335984141236149260","9622487279722262400","1106321683121913960","72473678497316448","1853020188851841"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["880","23716","617584","15695152","391676032","9637291840","234469641472","5652214974208","135214152392704","3213750657762304","75961379183472640","1786845365715841024","41855888566696247296","976825939928167825408","22722091290558400626688"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["403","396","81"]],"recurrence":{"denominator":{"coeffs":[["22","9"],["9","0"]]},"terms":[{"coeff":{"coeffs":[["403","396","81"],["117","81","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193595,"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-3148cd717b80282d"]},{"id":193596,"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-3148cd717b80282d"]},{"id":193597,"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-3148cd717b80282d"]},{"id":193598,"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-3148cd717b80282d"]}],"region_exact":{},"incoming_edges":[{"id":"e-3148cd717b80282d","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-599179c3856533fe","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":"502f9ffa75e1c30e7c54033c016f36d670db2a89681b729e5d052f6e582e6089","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-7e380db7ce29966b","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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:            403,             396,              81;\n  1:           6760,           10881,            5346,             729;\n  2:         107653,          243360,          195858,           64152,            6561;\n  3:        1656538,         4844385,         5475600,         2937870,          721710,           59049;\n  4:       24876631,        89453052,       130798395,        98560800,        39661245,         7794468,          531441;\n  5:      366837484,      1567227753,      2817771138,      2746766295,      1552332600,       499731687,        81841914,         4782969;\n  6:     5333623945,     26412298848,     56420199108,     67626507312,     49441793310,     22353589440,      5996780244,       841802544,       43046721;\n  7:    76678687774,    432023539545,   1069698103344,   1523345375916,   1369436773068,    800957051622,    301773457440,     69391314252,     8523250758,     387420489;\n  8:  1092263435419,   6901081899660,  19441059279525,  32090943100320,  34275270958110,  24649861915224,  12014355774330,   3879944452800,   780652285335,   85232507580,   3486784401;\n  9: 15440151087088, 108134080106481, 341603554033170, 641554956224325, 794250841732920, 678650364970578, 406722721601196, 169917317379810, 48014312603400, 8587175138685, 843801825042, 31381059609;\n","recurrence_sentence":"The row polynomials satisfy (9*n+22+9*x)*P(n,x) = (117*n+403+(81*n+396)*x+81*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 403 + 396*x + 81*x^2.","mathematica":"P[0, x_] := 403 + 396*x + 81*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((117*n + 403 + (81*n + 396)*x + 81*x^2)*P[n - 1, x])/(9*n + 22 + 9*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 403 + 396*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((117*n + 403 + (81*n + 396)*x + 81*x^2)*v[i - 1])/(9*n + 22 + 9*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-4397e742863fdbe2","steps":["basis A013610","D+1"],"edges":["e-c80be26a429cc3fc","e-3148cd717b80282d"],"text":"A038305 \u2192 basis A013610 \u2192 D+1"},"headline":"A038305 \u2192 basis A013610 \u2192 D+1","chain":null}