{"id":"n-507d474f4c2321e5","revision":1,"first_index":0,"fingerprint":"f6-6052c7b42ebfb2ec","name":"derivative of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-2,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":true,"rows":[[],[],["2"],["4"],["8","8"],["16","16"],["24","48","24"],["48","96","48"],["64","192","192","64"],["128","384","384","128"],["160","640","960","640","160"],["320","1280","1920","1280","320"],["384","1920","3840","3840","1920","384"],["768","3840","7680","7680","3840","768"],["896","5376","13440","17920","13440","5376","896"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","2","4","16","32","96","192","512","1024","2560","5120","12288","24576","57344"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-16","0"],["9","-3"],["-1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-8","-8","0"],["2","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161095,"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-5c70b541165ff56b"]},{"id":161096,"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-5c70b541165ff56b"]},{"id":161097,"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-5c70b541165ff56b"]}],"region_exact":{},"incoming_edges":[{"id":"e-5c70b541165ff56b","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-507d474f4c2321e5","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":"868da22d8689c12c9fa9fc2459c93c66ef89828f0c506f0d35e381a5eac2aed1","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-b6aca576a689e9f0","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","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-21e47ec0fe8e6c8a","kind":"fingerprint_related","node_a_id":"n-507d474f4c2321e5","node_b_id":"n-b7506a83efd6007f","fingerprint":"f6-6052c7b42ebfb2ec","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6052c7b42ebfb2ec"},"other_id":"n-b7506a83efd6007f","other":{"id":"n-b7506a83efd6007f","name":"basis_map_A013609 of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-6052c7b42ebfb2ec","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["2","2"]],"card_rows":[["1"],["1"],["2","2"],["2","2"]],"row_sums_preview":["1","1","4","4","16","16","64","64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["basis A013609"],"edges":["e-500dd9b78517f374"],"text":"A152198 \u2192 basis A013609"},"headline":"A152198 \u2192 basis A013609","chain":null}},{"id":"l-39af508258a6f80d","kind":"fingerprint_related","node_a_id":"n-4124f4242e7d4c67","node_b_id":"n-507d474f4c2321e5","fingerprint":"f6-6052c7b42ebfb2ec","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6052c7b42ebfb2ec"},"other_id":"n-4124f4242e7d4c67","other":{"id":"n-4124f4242e7d4c67","name":"Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-6052c7b42ebfb2ec","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","1"]],"row_sums_preview":["1","1","2","2","4","4","8","8"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A152198","A152815"],"derivation":{"root":"n-4124f4242e7d4c67","steps":[],"edges":[],"text":"A152198"},"headline":"A152198: Triangle read by rows, A007318 rows repeated.","chain":null}},{"id":"l-5d8fe8762d8e6ecb","kind":"fingerprint_related","node_a_id":"n-b6aca576a689e9f0","node_b_id":"n-507d474f4c2321e5","fingerprint":"f6-6052c7b42ebfb2ec","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6052c7b42ebfb2ec"},"other_id":"n-b6aca576a689e9f0","other":{"id":"n-b6aca576a689e9f0","name":"Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","first_index":0,"fingerprint":"f6-6052c7b42ebfb2ec","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["2","2"]],"card_rows":[["1"],["2"],["2","2"],["4","4"]],"row_sums_preview":["1","2","4","8","16","32","64","128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A274496"],"derivation":{"root":"n-b6aca576a689e9f0","steps":[],"edges":[],"text":"A274496"},"headline":"A274496: Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","chain":null}},{"id":"l-ba9eac41170708b2","kind":"fingerprint_related","node_a_id":"n-688648772a3128e6","node_b_id":"n-507d474f4c2321e5","fingerprint":"f6-6052c7b42ebfb2ec","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6052c7b42ebfb2ec"},"other_id":"n-688648772a3128e6","other":{"id":"n-688648772a3128e6","name":"derivative of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-6052c7b42ebfb2ec","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["1"]],"card_rows":[["1"],["1"],["2","2"],["2","2"]],"row_sums_preview":["0","0","1","1","4","4","12","12"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["D"],"edges":["e-598ca892414d6ed7"],"text":"A152198 \u2192 D"},"headline":"A152198 \u2192 D","chain":null}},{"id":"l-d5c2ad6fbcd8e5ea","kind":"fingerprint_related","node_a_id":"n-507d474f4c2321e5","node_b_id":"n-2b301932743053f0","fingerprint":"f6-6052c7b42ebfb2ec","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6052c7b42ebfb2ec"},"other_id":"n-2b301932743053f0","other":{"id":"n-2b301932743053f0","name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","first_index":0,"fingerprint":"f6-6052c7b42ebfb2ec","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["4","4"]],"card_rows":[["1"],["2"],["4","4"],["8","8"]],"row_sums_preview":["1","2","8","16","64","128","512","1024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b6aca576a689e9f0","steps":["basis A013609"],"edges":["e-267456e89d3343d1"],"text":"A274496 \u2192 basis A013609"},"headline":"A274496 \u2192 basis A013609","chain":null}},{"id":"l-f36aa2ffa723413d","kind":"fingerprint_related","node_a_id":"n-35bd700768cac55f","node_b_id":"n-507d474f4c2321e5","fingerprint":"f6-6052c7b42ebfb2ec","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6052c7b42ebfb2ec"},"other_id":"n-35bd700768cac55f","other":{"id":"n-35bd700768cac55f","name":"Triangle read by rows: triangle of repeated stepped binomial coefficients.","first_index":0,"fingerprint":"f6-6052c7b42ebfb2ec","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","1"],["0","1","1"]],"card_rows":[["1"],["0","1"],["0","1","1"],["0","0","1","1"]],"row_sums_preview":["1","1","2","2","4","4","8","8"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A103633"],"derivation":{"root":"n-35bd700768cac55f","steps":[],"edges":[],"text":"A103633"},"headline":"A103633: Triangle read by rows: triangle of repeated stepped binomial coefficients.","chain":null}},{"id":"l-4479e9dd924a6cf6","kind":"oeis_match","node_a_id":"n-507d474f4c2321e5","node_b_id":null,"fingerprint":null,"oeis_id":"A274497","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":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A274497","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A274497","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-507d474f4c2321e5","oeis_id":"A274497","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":31,"oeis_terms_available":31,"term_count":200,"raw_size":7235}],"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:   0;\n  2:   2;\n  3:   4;\n  4:   8,   8;\n  5:  16,  16;\n  6:  24,  48,  24;\n  7:  48,  96,  48;\n  8:  64, 192, 192,  64;\n  9: 128, 384, 384, 128;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(n^2-9*n+16-(n^2-3*n)*x)*P(n-1,x) + (2*n-8-8*x-2*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 9*n + 16 - (n^2 - 3*n)*x)*P[n - 1, x] + (2*n - 8 - 8*x - 2*n*x^2)*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = (-(n^2 - 9*n + 16 - (n^2 - 3*n)*x)*v[i - 1] + (2*n - 8 - 8*x - 2*n*x^2)*deriv(v[i - 1], x))/(n - 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-b6aca576a689e9f0","steps":["D"],"edges":["e-5c70b541165ff56b"],"text":"A274496 \u2192 D"},"headline":"A274496 \u2192 D","chain":null}
