{"id":"n-e70bc71f808d87be","revision":1,"first_index":0,"fingerprint":"f6-4f3de3e72d370144","name":"one_plus_derivative of Triangular array formed by the Mersenne numbers.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","1"],["13","8","1"],["38","41","12","1"],["103","168","84","16","1"],["264","597","444","142","20","1"],["649","1920","1959","920","215","24","1"],["1546","5737","7608","4855","1650","303","28","1"],["3595","16208","26866","22128","10140","2688","406","32","1"],["8204","43829","88152","90308","53088","18858","4088","524","36","1"],["18445","114456","272877","338096","246050","111552","32242","5904","657","40","1"],["40974","290553","805812","1181165","1035940","579026","212688","51714","8190","805","44","1"],["90127","720600","2289072","3899440","4035595","2709360","1222536","376416","78885","11000","968","48","1"],["196624","1752757","6295524","12281642","14745060","11650599","6295656","2373852","628056","115555","14388","1146","52","1"],["426001","4193936","16847443","37177992","51056937","46704888","29629803","13344144","4312011","998976","163713","18408","1339","56","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","22","92","372","1468","5688","21728","82064","307088","1140320","4206912","15434048","56350912","204875648"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["14","-6","-2"],["0","0","0"],["2","0","0"]]},"terms":[{"coeff":{"coeffs":[["23","-7","-7","-1"],["29","-5","-7","-1"],["9","3","0","0"],["3","1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","7","-29","-11","-1"],["1","6","1","0","0"],["1","6","1","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164281,"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-9b8106d62ea669b5"]},{"id":164282,"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-9b8106d62ea669b5"]},{"id":164283,"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-9b8106d62ea669b5"]},{"id":164284,"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-9b8106d62ea669b5"]}],"region_exact":{},"incoming_edges":[{"id":"e-9b8106d62ea669b5","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-e70bc71f808d87be","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":"3c9518423cf1dc801889f9a3996cb5b0c4ba554b08e32f713849128aa98e0f63","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-9b5c7e97e5191da6","role":"sequence","name":"Triangular array formed by the Mersenne numbers.","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":[{"id":"l-71f608fec87947fb","kind":"oeis_match","node_a_id":"n-e70bc71f808d87be","node_b_id":null,"fingerprint":null,"oeis_id":"A053297","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A053297","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A053297","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-e70bc71f808d87be","oeis_id":"A053297","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":26,"oeis_terms_available":26,"term_count":200,"raw_size":11908}],"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:    1;\n  1:    4,     1;\n  2:   13,     8,     1;\n  3:   38,    41,    12,     1;\n  4:  103,   168,    84,    16,     1;\n  5:  264,   597,   444,   142,    20,     1;\n  6:  649,  1920,  1959,   920,   215,    24,    1;\n  7: 1546,  5737,  7608,  4855,  1650,   303,   28,   1;\n  8: 3595, 16208, 26866, 22128, 10140,  2688,  406,  32,  1;\n  9: 8204, 43829, 88152, 90308, 53088, 18858, 4088, 524, 36, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+14*n-6*n*x-2*n*x^2)*P(n,x) = (3*n^3+9*n^2+29*n+23+(n^3+3*n^2-5*n-7)*x-(7*n+7)*x^2-(n+1)*x^3)*P(n-1,x) + (n^2+n+2+(6*n^2+6*n+7)*x+(n^2+n-29)*x^2-11*x^3-x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^3 + 9*n^2 + 29*n + 23 + (n^3 + 3*n^2 - 5*n - 7)*x - (7*n + 7)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (n^2 + n + 2 + (6*n^2 + 6*n + 7)*x + (n^2 + n - 29)*x^2 - 11*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 + 14*n - 6*n*x - 2*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^3 + 9*n^2 + 29*n + 23 + (n^3 + 3*n^2 - 5*n - 7)*x - (7*n + 7)*x^2 - (n + 1)*x^3)*v[i - 1] + (n^2 + n + 2 + (6*n^2 + 6*n + 7)*x + (n^2 + n - 29)*x^2 - 11*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 + 14*n - 6*n*x - 2*n*x^2)); 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-9b5c7e97e5191da6","steps":["D+1"],"edges":["e-9b8106d62ea669b5"],"text":"A110441 \u2192 D+1"},"headline":"A110441 \u2192 D+1","chain":null}
