{"id":"n-3234ee4840ebe620","revision":1,"first_index":1,"fingerprint":"f6-daaec9674abe365e","name":"derivative of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","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":false,"rows":[[],["1"],["3","2"],["5","10","3"],["7","26","21","4"],["9","50","75","36","5"],["11","82","189","164","55","6"],["13","122","387","516","305","78","7"],["15","170","693","1284","1155","510","105","8"],["17","226","1131","2724","3405","2262","791","136","9"],["19","290","1725","5156","8415","7734","4025","1160","171","10"],["21","362","2499","8964","18265","21918","15687","6664","1629","210","11"],["23","442","3477","14596","35915","53934","50281","29192","10431","2210","253","12"],["25","530","4683","22564","65365","118950","138775","104584","50769","15610","2915","300","13"],["27","626","6141","33444","111815","240486","340473","320648","201267","83610","22517","3756","351","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","5","18","58","175","507","1428","3940","10701","28705","76230","200766","525083","1365175"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-10","0","2"],["9","-4","-1"],["-2","2","0"]]},"terms":[{"coeff":{"coeffs":[["-6","-38","-18","-2"],["7","36","9","0"],["-2","-6","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-19","-9","-1"],["0","2","12","2","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161425,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-e701479dc0b370a7"]},{"id":161426,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-e701479dc0b370a7"]},{"id":161427,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-e701479dc0b370a7"]},{"id":161428,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-e701479dc0b370a7"]}],"region_exact":{},"incoming_edges":[{"id":"e-e701479dc0b370a7","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-3234ee4840ebe620","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":"277e82df6341b785b5e48fc249958bc5fb395c52a7f2639ca79d3c21569e3ca0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-811c78acf96cbc19","role":"sequence","name":"Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-fa9d8a3e12fc4aff","kind":"fingerprint_related","node_a_id":"n-3234ee4840ebe620","node_b_id":"n-edd7a038474f2f4a","fingerprint":"f6-daaec9674abe365e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-daaec9674abe365e"},"other_id":"n-edd7a038474f2f4a","other":{"id":"n-edd7a038474f2f4a","name":"derivative of delannoy","first_index":0,"fingerprint":"f6-daaec9674abe365e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["3","2"]],"card_rows":[["1"],["3","2"],["5","10","3"],["7","26","21","4"]],"row_sums_preview":["0","1","5","18","58","175","507","1428"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e00a1cb428776eaa","steps":["D"],"edges":["e-2037b073b28f1db1"],"text":"delannoy \u2192 D"},"headline":"delannoy \u2192 D","chain":null}},{"id":"l-233ca14e372d6d96","kind":"oeis_match","node_a_id":"n-3234ee4840ebe620","node_b_id":null,"fingerprint":null,"oeis_id":"A364553","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A364553","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A364553","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-3234ee4840ebe620","oeis_id":"A364553","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":30,"oeis_terms_available":30,"term_count":200,"raw_size":8906}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  0;\n   2:  1;\n   3:  3,   2;\n   4:  5,  10,    3;\n   5:  7,  26,   21,    4;\n   6:  9,  50,   75,   36,    5;\n   7: 11,  82,  189,  164,   55,    6;\n   8: 13, 122,  387,  516,  305,   78,   7;\n   9: 15, 170,  693, 1284, 1155,  510, 105,   8;\n  10: 17, 226, 1131, 2724, 3405, 2262, 791, 136, 9;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^2-9*n+10-(2*n^2-4*n)*x+(n-2)*x^2)*P(n,x) = -(2*n^2-7*n+6+(6*n^2-36*n+38)*x-(9*n-18)*x^2+2*x^3)*P(n-1,x) + ((2*n-3)*x+(12*n-19)*x^2+(2*n-9)*x^3-x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 7*n + 6 + (6*n^2 - 36*n + 38)*x - (9*n - 18)*x^2 + 2*x^3)*P[n - 1, x] + ((2*n - 3)*x + (12*n - 19)*x^2 + (2*n - 9)*x^3 - x^4)*D[P[n - 1, x], x])/(-2*n^2 + 9*n - 10 + (2*n^2 - 4*n)*x - (n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 7*n + 6 + (6*n^2 - 36*n + 38)*x - (9*n - 18)*x^2 + 2*x^3)*v[i - 1] + ((2*n - 3)*x + (12*n - 19)*x^2 + (2*n - 9)*x^3 - x^4)*deriv(v[i - 1], x))/(-2*n^2 + 9*n - 10 + (2*n^2 - 4*n)*x - (n - 2)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-811c78acf96cbc19","steps":["D"],"edges":["e-e701479dc0b370a7"],"text":"A008288 \u2192 D"},"headline":"A008288 \u2192 D","chain":null}
