{"id":"n-05641e9f16c28d22","revision":0,"first_index":0,"fingerprint":"f6-a686d7a4a52344cd","name":"derivative of basis_map_A007318 of delannoy","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":[[],["1"],["5","2"],["18","16","3"],["58","80","33","4"],["175","328","213","56","5"],["507","1202","1071","444","85","6"],["1428","4096","4650","2656","800","120","7"],["3940","13264","18318","13384","5560","1308","161","8"],["10701","41360","67326","60048","31970","10368","1995","208","9"],["28705","125266","234906","247672","161290","67080","17773","2888","261","10"],["76230","370704","787077","958368","739200","376440","127897","28576","4014","320","11"],["200766","1076544","2553015","3527052","3147240","1900548","791007","226520","43686","5400","385","12"],["525083","3077784","8063979","12465928","12640455","8841264","4366397","1531792","378369","64120","7073","456","13"],["1365175","8683810","24912465","42614900","48419375","38528310","22055915","9184280","2778525","602590","91003","9060","533","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","7","37","175","777","3315","13757","55943","223985","885851","3468837","13472175","51962713","199245955"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-4","0","1"],["4","-2","-1"],["0","2","0"]]},"terms":[{"coeff":{"coeffs":[["-20","-32","-15","-2"],["36","42","9","0"],["-8","-6","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-16","-40","-34","-11","-1"],["16","32","18","2","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174735,"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-bc97082752bc979e"]},{"id":174736,"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-bc97082752bc979e"]},{"id":174737,"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-bc97082752bc979e"]}],"region_exact":{},"incoming_edges":[{"id":"e-bc97082752bc979e","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-05641e9f16c28d22","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":"e74d6f64f704cf4548ff91e7023083fc66134aea758eaf4c8d0b7e12d943c370","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-2a15a23ed2e4e7bb","role":"sequence","name":"basis_map_A007318 of delannoy","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-d295013e9997b968","kind":"fingerprint_related","node_a_id":"n-cc1766317c70b9d1","node_b_id":"n-05641e9f16c28d22","fingerprint":"f6-a686d7a4a52344cd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a686d7a4a52344cd"},"other_id":"n-cc1766317c70b9d1","other":{"id":"n-cc1766317c70b9d1","name":"derivative of shift_x_plus_one of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","first_index":1,"fingerprint":"f6-a686d7a4a52344cd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["5","2"]],"card_rows":[["1"],["5","2"],["18","16","3"],["58","80","33","4"]],"row_sums_preview":["0","1","7","37","175","777","3315","13757"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-811c78acf96cbc19","steps":["basis A007318","D"],"edges":["e-0985ee5bc77b64e0","e-83df4345d7e0e4cb"],"text":"A008288 \u2192 basis A007318 \u2192 D"},"headline":"A008288 \u2192 basis A007318 \u2192 D","chain":null}}],"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:     1;\n  2:     5,     2;\n  3:    18,    16,     3;\n  4:    58,    80,    33,     4;\n  5:   175,   328,   213,    56,     5;\n  6:   507,  1202,  1071,   444,    85,     6;\n  7:  1428,  4096,  4650,  2656,   800,   120,    7;\n  8:  3940, 13264, 18318, 13384,  5560,  1308,  161,   8;\n  9: 10701, 41360, 67326, 60048, 31970, 10368, 1995, 208, 9;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4+(2*n^2-2*n)*x-(n-1)*x^2)*P(n,x) = -(8*n^2-36*n+20+(6*n^2-42*n+32)*x-(9*n-15)*x^2+2*x^3)*P(n-1,x) + (16*n-16+(32*n-40)*x+(18*n-34)*x^2+(2*n-11)*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 - 36*n + 20 + (6*n^2 - 42*n + 32)*x - (9*n - 15)*x^2 + 2*x^3)*P[n - 1, x] + (16*n - 16 + (32*n - 40)*x + (18*n - 34)*x^2 + (2*n - 11)*x^3 - x^4)*D[P[n - 1, x], x])/(4*n - 4 + (2*n^2 - 2*n)*x - (n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^2 - 36*n + 20 + (6*n^2 - 42*n + 32)*x - (9*n - 15)*x^2 + 2*x^3)*v[i - 1] + (16*n - 16 + (32*n - 40)*x + (18*n - 34)*x^2 + (2*n - 11)*x^3 - x^4)*deriv(v[i - 1], x))/(4*n - 4 + (2*n^2 - 2*n)*x - (n - 1)*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-e00a1cb428776eaa","steps":["x \u21a6 x+1","D"],"edges":["e-41d30650643fabff","e-bc97082752bc979e"],"text":"delannoy \u2192 x \u21a6 x+1 \u2192 D"},"headline":"delannoy \u2192 x \u21a6 x+1 \u2192 D","chain":null}