{"id":"n-22d01599550a0f21","revision":0,"first_index":0,"fingerprint":"f6-b643eea373acac66","name":"derivative of basis_map_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k=0 and 0<=k<=2n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["18","18"],["216","864","972","324"],["1944","13608","34992","40824","21870","4374"],["15552","155520","629856","1329696","1574640","1049760","367416","52488"],["116640","1516320","8398080","25894080","48813840","58261680","44089920","20470320","5314410","590490"],["839808","13436928","94478400","384212160","1001471040","1753519104","2103089184","1728954720","956593800","340122240","70150212","6377292"],["5878656","111694464","952342272","4814619264","16070775840","37339753248","62060971392","74679506496","65086642152","40623350040","17677853424","5089079016","870500358","66961566"],["40310784","886837248","8888527872","53754430464","219038722560","636164637696","1358992422144","2172836918016","2620913956992","2385617391360","1626362515008","816395412672","292488120288","70711413696","10331213040","688747536"],["272097792","6802444800","78364164096","551814322176","2657034938880","9273909044736","24273299828736","48627412701696","75450915073728","91176398815680","85823452965888","62598886051968","34973222383008","14666190031584","4463084033280","929809173600","118550669634","6973568802"],["1813985280","50791587840","661197634560","5314069877760","29533494343680","120484902297600","373870529141760","902387838366720","1718287352812800","2603135087262720","3150193480156800","3045558954487680","2343367066584960","1423227908390400","672809918016960","242122308805440","64017361602360","11715595587360","1324978072380","69735688020"],["11972302848","371141388288","5387536281600","48667411077120","306685502830080","1432572834958848","5145528151830528","14551196742973440","32909293966728960","60136420761457920","89344396489683456","108245557370624256","106955205391869120","85943005763186880","55819792004232960","29009599907916672","11877354489291192","3741877547777160","874485527770800","142679217688920","14498049539358","690383311398"],["78364164096","2664381579264","42669287350272","427985882210304","3015942810439680","15876853920423936","64821435780777984","210348919315003392","551658609316767744","1182705367261777920","2088769743648986112","3053345746790366208","3702819091014111744","3725521906874600448","3103079677406818560","2129782808064409344","1195434603617829696","542541867561986688","195814799378437536","54844050257457120","11482455235171536","1690058346302304","155901104137512","6778308875544"],["509367066624","18846581465088","330069859172352","3639937058095104","28365378522624000","166183297637879808","760343426425110528","2786085298996826112","8319593886064141824","20495829732199810560","42018429436161564672","72109134686094944256","103974071154581481984","126190985700666152448","128920181224586618880","110677480553878977024","79556116535488356192","47602879288328583648","23507849998692423936","9464658123282373440","3053289232988794800","769622746347016848","145923433472711232","19562199414819984","1652212788413850","66088511536554"],["3291294892032","131651795681280","2503029765390336","30100537435078656","257041902830469120","1658931112200241152","8409313578331201536","34352061852094930944","115134050269704001536","320710315728195993600","749443538061231197184","1479098929061681980416","2476895848218240583680","3529789207712188710912","4286935121916185625600","4437296787185045941248","3909029363296649085312","2922472752073185991680","1845742743386192274624","978228636335047056960","431043018651804791232","155894163149223442944","45455149526749548768","10414492001976325536","1804216364947924200","222057398762821440","17294855095950516","640550188738908"]],"degrees":[-1,1,3,5,7,9,11,13,15,17,19,21,23,25,27],"row_sums":["0","36","2376","117612","5174928","213465780","8453244888","325449928188","12274111577376","455676392310084","16708134384703080","606505278164721804","21834190013929984944","780572292997996961748","27740338412698045871352"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["18","18"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["6","18","9"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192682,"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-86043a1b9810de3d"]},{"id":192683,"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-86043a1b9810de3d"]},{"id":192684,"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-86043a1b9810de3d"]}],"region_exact":{},"incoming_edges":[{"id":"e-86043a1b9810de3d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-22d01599550a0f21","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":"17e4b964b53e8b57ed8cd89f76b55b3bd3295f5b0f5a54cf1564be4fe13fd7d6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c4fd821a14172d5f","role":"sequence","name":"basis_map_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k=0 and 0<=k<=2n.","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: 0;\n 1: 18, 18;\n 2: 216, 864, 972, 324;\n 3: 1944, 13608, 34992, 40824, 21870, 4374;\n 4: 15552, 155520, 629856, 1329696, 1574640, 1049760, 367416, 52488;\n 5: 116640, 1516320, 8398080, 25894080, 48813840, 58261680, 44089920, 20470320, 5314410, 590490;\n 6: 839808, 13436928, 94478400, 384212160, 1001471040, 1753519104, 2103089184, 1728954720, 956593800, 340122240, 70150212, 6377292;\n 7: 5878656, 111694464, 952342272, 4814619264, 16070775840, 37339753248, 62060971392, 74679506496, 65086642152, 40623350040, 17677853424, 5089079016, 870500358, 66961566;\n 8: 40310784, 886837248, 8888527872, 53754430464, 219038722560, 636164637696, 1358992422144, 2172836918016, 2620913956992, 2385617391360, 1626362515008, 816395412672, 292488120288, 70711413696, 10331213040, 688747536;\n 9: 272097792, 6802444800, 78364164096, 551814322176, 2657034938880, 9273909044736, 24273299828736, 48627412701696, 75450915073728, 91176398815680, 85823452965888, 62598886051968, 34973222383008, 14666190031584, 4463084033280, 929809173600, 118550669634, 6973568802;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (6*n+18*n*x+9*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 18 + 18*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 18 + 18*x;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 18*n*x + 9*n*x^2)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 18 + 18*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 18*n*x + 9*n*x^2)*v[i - 1])/(n - 1)); 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-d3c3090977105807","steps":["basis A013610","D"],"edges":["e-0f2bf5edb3121755","e-86043a1b9810de3d"],"text":"A272867 \u2192 basis A013610 \u2192 D"},"headline":"A272867 \u2192 basis A013610 \u2192 D","chain":null}