{"id":"n-7938474c51929009","revision":0,"first_index":0,"fingerprint":"f6-9783f43730ed991f","name":"derivative of basis_map_A013609 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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":[[],["2"],["22","16"],["244","376","144"],["3036","7288","5808","1536"],["42846","140888","173208","94336","19200"],["681738","2859848","4787688","3997696","1664640","276480"],["12117960","61986384","131872128","149335616","94929280","32113152","4515840"],["238437720","1441543248","3729499392","5351977536","4600510080","2368565760","676214784","82575360"],["5150586330","35973079632","109782156144","191195558592","207829186080","144373767168","62587961856","15479635968","1672151040"],["121256894430","961458125472","3384617667984","6942630894272","9144301872480","8019749576448","4683068432896","1755654340608","383406981120","37158912000"],["3091393404540","27448840240776","109574324504208","258962628829120","401241292659520","425863809691776","313550813085440","158125770432512","52270805495808","10226825625600","899245670400"],["84874454550900","834607889437320","3727516361273136","9980444104862656","17799911181361600","22202344470888576","19762956998505728","12553395134013440","5576165939073024","1649558787194880","292469338275840","23543886643200"],["2497162465970550","26947884651454440","133192649442430152","398692558022093760","804962035304799520","1154820529572473472","1207070954576608128","926183819431104512","517743288024864768","205627325494149120","55074053447221248","8931178897735680","663152807116800"],["78394536820098450","921329752986529560","4994397989191560888","16534609761824778048","37298138554612467680","60536519743842994560","72729194982089102464","65485601309953490944","44189345664800870400","22070761400103895040","7930469175189209088","1941266184240365568","290143857971036160","19996607722291200"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","38","764","17668","470478","14268090","486870360","18489323880","774044082810","35433303697710","1761255749639700","94487788546080300","5442406592138022150","335020169521348690050"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["-1","-4","-4"],["4","12","8"],["-4","-8","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","0"],["-2","-4"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189221,"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-8defcd99daf41f2a"]},{"id":189222,"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-8defcd99daf41f2a"]},{"id":189223,"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-8defcd99daf41f2a"]},{"id":189224,"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-8defcd99daf41f2a"]}],"region_exact":{},"incoming_edges":[{"id":"e-8defcd99daf41f2a","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-7938474c51929009","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":"e920b332982d250fa1a237d3700a6c186df166bcaada7ca1b00da96161b0bc6e","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-8401ba73e6a78cca","role":"sequence","name":"basis_map_A013609 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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:          2;\n  2:         22,          16;\n  3:        244,         376,          144;\n  4:       3036,        7288,         5808,         1536;\n  5:      42846,      140888,       173208,        94336,        19200;\n  6:     681738,     2859848,      4787688,      3997696,      1664640,       276480;\n  7:   12117960,    61986384,    131872128,    149335616,     94929280,     32113152,     4515840;\n  8:  238437720,  1441543248,   3729499392,   5351977536,   4600510080,   2368565760,   676214784,    82575360;\n  9: 5150586330, 35973079632, 109782156144, 191195558592, 207829186080, 144373767168, 62587961856, 15479635968, 1672151040;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(8*n^3-12*n^2+4*n)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(4*n^2-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (8*n^3 - 12*n^2 + 4*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (4*n^2 - 4*n)*x)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (8*n^3 - 12*n^2 + 4*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (4*n^2 - 4*n)*x)*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-7fd16a32c438d522","steps":["basis A013609","D"],"edges":["e-23e30432bbaed1be","e-8defcd99daf41f2a"],"text":"A059364 \u2192 basis A013609 \u2192 D"},"headline":"A059364 \u2192 basis A013609 \u2192 D","chain":null}