{"id":"n-4e34a7abc01c81bb","revision":1,"first_index":0,"fingerprint":"f6-019617c6f1a48892","name":"veronese_section_2_1 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["4"],["12","1"],["32","8"],["80","40","1"],["192","160","12"],["448","560","84","1"],["1024","1792","448","16"],["2304","5376","2016","144","1"],["5120","15360","8064","960","20"],["11264","42240","29568","5280","220","1"],["24576","112640","101376","25344","1760","24"],["53248","292864","329472","109824","11440","312","1"],["114688","745472","1025024","439296","64064","2912","28"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","4","13","40","121","364","1093","3280","9841","29524","88573","265720","797161","2391484"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_1","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163287,"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-868a2ac5701aa5c6"]},{"id":163288,"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-868a2ac5701aa5c6"]},{"id":163289,"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-868a2ac5701aa5c6"]}],"region_exact":{},"incoming_edges":[{"id":"e-868a2ac5701aa5c6","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-4e34a7abc01c81bb","premise_count":1,"theorem":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"3f804017eb5858276767b174c9854f8864979033b70df44aad4f70466b24b727","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-af754f07daf65b5e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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-ca8a2b6127db909e","kind":"fingerprint_related","node_a_id":"n-4e34a7abc01c81bb","node_b_id":"n-a1bd1d1e7fcaa1ad","fingerprint":"f6-019617c6f1a48892","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-019617c6f1a48892"},"other_id":"n-a1bd1d1e7fcaa1ad","other":{"id":"n-a1bd1d1e7fcaa1ad","name":"veronese_section_2_1 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-019617c6f1a48892","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["3"],["12"]],"card_rows":[["3"],["12"],["36","3"],["96","24"]],"row_sums_preview":["0","3","12","39","120","363","1092","3279"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["Veronese 2 (j=1)"],"edges":["e-7d561377a0a291f9"],"text":"A120910 \u2192 Veronese 2 (j=1)"},"headline":"A120910 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-ea9fb6970166eee2","kind":"oeis_match","node_a_id":"n-4e34a7abc01c81bb","node_b_id":null,"fingerprint":null,"oeis_id":"A261547","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":1,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A261547","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f43274eb37b643f7","kind":"oeis_match","node_a_id":"n-4e34a7abc01c81bb","node_b_id":null,"fingerprint":null,"oeis_id":"A003462","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":"A003462","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A003462","relation":"match_row_sums"},{"oeis_id":"A261547","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-4e34a7abc01c81bb","oeis_id":"A003462","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":27,"oeis_terms_available":27,"term_count":200,"raw_size":10425}],"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:    4;\n  3:   12,    1;\n  4:   32,    8;\n  5:   80,   40,    1;\n  6:  192,  160,   12;\n  7:  448,  560,   84,   1;\n  8: 1024, 1792,  448,  16;\n  9: 2304, 5376, 2016, 144, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 4*P(n-1,x) - (4-x)*P(n-2,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[4*P[n - 1, x] - (4 - x)*P[n - 2, x]];\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] = 4*v[i - 1] - (4 - x)*v[i - 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-af754f07daf65b5e","steps":["Veronese 2 (j=1)"],"edges":["e-868a2ac5701aa5c6"],"text":"A038207 \u2192 Veronese 2 (j=1)"},"headline":"A038207 \u2192 Veronese 2 (j=1)","chain":null}