{"id":"n-d321ff0d3bb39024","revision":1,"first_index":0,"fingerprint":"f6-98cc1557e904900a","name":"shift_x_plus_one 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":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["2"],["4","2"],["8","4"],["16","16","4"],["32","32","8"],["64","96","48","8"],["128","192","96","16"],["256","512","384","128","16"],["512","1024","768","256","32"],["1024","2560","2560","1280","320","32"],["2048","5120","5120","2560","640","64"],["4096","12288","15360","10240","3840","768","64"],["8192","24576","30720","20480","7680","1536","128"],["16384","57344","86016","71680","35840","10752","1792","128"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","6","12","36","72","216","432","1296","2592","7776","15552","46656","93312","279936"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","-1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-706a96c1242f8bb9","recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161107,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-5428ec8f88838d6b"]},{"id":161108,"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-5428ec8f88838d6b","e-9477b3c3a3760be1"]},{"id":161109,"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-5428ec8f88838d6b","e-9477b3c3a3760be1"]},{"id":161110,"position":3,"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-9477b3c3a3760be1"]}],"region_exact":{},"incoming_edges":[{"id":"e-9477b3c3a3760be1","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-d321ff0d3bb39024","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"9d002a92faa5578a8a83ab2fb821c87c198a1bb9e75885e13a3d08ac8029d157","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]"}]},{"id":"e-706a96c1242f8bb9","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-d321ff0d3bb39024","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"00fe99fd95d38dfca83405203eb80715042ac3738bbef305426d30e50b944d36","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-5428ec8f88838d6b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-d321ff0d3bb39024","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"3c8f1aac7e203cac27ea2d8acdb45777f3f296708b8c0d4bd5c73b73d94f4a4d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"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-03a36deca288fa35","kind":"fingerprint_related","node_a_id":"n-8c8ff2e102336282","node_b_id":"n-d321ff0d3bb39024","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-8c8ff2e102336282","other":{"id":"n-8c8ff2e102336282","name":"shift_x_plus_one of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["2","1"]],"row_sums_preview":["1","1","3","3","9","9","27","27"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["x \u21a6 x+1"],"edges":["e-009b68c8336dc2e0"],"text":"A152198 \u2192 x \u21a6 x+1"},"headline":"A152198 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-1e87fbe2c8df0724","kind":"fingerprint_related","node_a_id":"n-d321ff0d3bb39024","node_b_id":"n-f42b095ef0d54419","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-f42b095ef0d54419","other":{"id":"n-f42b095ef0d54419","name":"derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["6"]],"card_rows":[["6"],["18"],["36","72"],["108","216"]],"row_sums_preview":["0","0","6","18","108","324","1458","4374"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c641d81e21514ce","steps":["D"],"edges":["e-7b8439e885873d7d"],"text":"A274498 \u2192 D"},"headline":"A274498 \u2192 D","chain":null}},{"id":"l-3836f1d7cacd0b10","kind":"fingerprint_related","node_a_id":"n-7c641d81e21514ce","node_b_id":"n-d321ff0d3bb39024","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-7c641d81e21514ce","other":{"id":"n-7c641d81e21514ce","name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3"],["3","6"]],"card_rows":[["1"],["3"],["3","6"],["9","18"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A274498"],"derivation":{"root":"n-7c641d81e21514ce","steps":[],"edges":[],"text":"A274498"},"headline":"A274498: Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","chain":null}},{"id":"l-42eb7fff41453db6","kind":"fingerprint_related","node_a_id":"n-d321ff0d3bb39024","node_b_id":"n-4d941e7b1ccdb34d","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-4d941e7b1ccdb34d","other":{"id":"n-4d941e7b1ccdb34d","name":"basis_map_A013611 of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["2","4"]],"card_rows":[["1"],["1"],["2","4"],["2","4"]],"row_sums_preview":["1","1","6","6","36","36","216","216"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["basis A013611"],"edges":["e-0bf2c6340da57ed0"],"text":"A152198 \u2192 basis A013611"},"headline":"A152198 \u2192 basis A013611","chain":null}},{"id":"l-73385763dc25e758","kind":"fingerprint_related","node_a_id":"n-d321ff0d3bb39024","node_b_id":"n-98c21bd2a6c7c1d4","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-98c21bd2a6c7c1d4","other":{"id":"n-98c21bd2a6c7c1d4","name":"basis_map_A013611 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":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["4","8"]],"card_rows":[["1"],["2"],["4","8"],["8","16"]],"row_sums_preview":["1","2","12","24","144","288","1728","3456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b6aca576a689e9f0","steps":["basis A013611"],"edges":["e-6f6524a2909f4dfb"],"text":"A274496 \u2192 basis A013611"},"headline":"A274496 \u2192 basis A013611","chain":null}},{"id":"l-73b0b35f2f252e71","kind":"fingerprint_related","node_a_id":"n-b4a70761c59dc931","node_b_id":"n-d321ff0d3bb39024","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-b4a70761c59dc931","other":{"id":"n-b4a70761c59dc931","name":"mobius_x_over_x_plus_one of Triangle read by rows: triangle of repeated stepped binomial coefficients.","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","1"],["0","1","2"]],"card_rows":[["1"],["0","1"],["0","1","2"],["0","0","1","2"]],"row_sums_preview":["1","1","3","3","9","9","27","27"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-35bd700768cac55f","steps":["x \u21a6 x/(x+1)"],"edges":["e-3c0198f97249d373"],"text":"A103633 \u2192 x \u21a6 x/(x+1)"},"headline":"A103633 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-bc5e7d1214f9ccf1","kind":"fingerprint_related","node_a_id":"n-d321ff0d3bb39024","node_b_id":"n-7619704d59388f48","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-7619704d59388f48","other":{"id":"n-7619704d59388f48","name":"basis_map_A013610 of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["3","6"]],"card_rows":[["1"],["1"],["3","6"],["3","6"]],"row_sums_preview":["1","1","9","9","81","81","729","729"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba3bb18490b15e2e","steps":["basis A013610"],"edges":["e-e7473649d9984108"],"text":"A188440 \u2192 basis A013610"},"headline":"A188440 \u2192 basis A013610","chain":null}},{"id":"l-d3767c71d3a6bbf8","kind":"fingerprint_related","node_a_id":"n-ba3bb18490b15e2e","node_b_id":"n-d321ff0d3bb39024","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-ba3bb18490b15e2e","other":{"id":"n-ba3bb18490b15e2e","name":"Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["1","2"]],"card_rows":[["1"],["1"],["1","2"],["1","2"]],"row_sums_preview":["1","1","3","3","9","9","27","27"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A188440"],"derivation":{"root":"n-ba3bb18490b15e2e","steps":[],"edges":[],"text":"A188440"},"headline":"A188440: Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","chain":null}},{"id":"l-dbc6b41da8f45a3f","kind":"fingerprint_related","node_a_id":"n-b3c0d8fe47ca78c3","node_b_id":"n-d321ff0d3bb39024","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-b3c0d8fe47ca78c3","other":{"id":"n-b3c0d8fe47ca78c3","name":"derivative of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["2"],["4","8"],["4","8"]],"row_sums_preview":["0","0","2","2","12","12","54","54"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba3bb18490b15e2e","steps":["D"],"edges":["e-107b5f015c60ec70"],"text":"A188440 \u2192 D"},"headline":"A188440 \u2192 D","chain":null}},{"id":"l-fcd152fb6c04b1e6","kind":"fingerprint_related","node_a_id":"n-d321ff0d3bb39024","node_b_id":"n-2b030ef584988651","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-2b030ef584988651","other":{"id":"n-2b030ef584988651","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3"],["9","18"]],"card_rows":[["1"],["3"],["9","18"],["27","54"]],"row_sums_preview":["1","3","27","81","729","2187","19683","59049"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c641d81e21514ce","steps":["basis A013610"],"edges":["e-b7097430306b08af"],"text":"A274498 \u2192 basis A013610"},"headline":"A274498 \u2192 basis A013610","chain":null}},{"id":"l-fc0862b07111df98","kind":"oeis_match","node_a_id":"n-d321ff0d3bb39024","node_b_id":null,"fingerprint":null,"oeis_id":"A026549","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":"A026549","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A026549","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-d321ff0d3bb39024","oeis_id":"A026549","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":32,"oeis_terms_available":32,"term_count":200,"raw_size":8726}],"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:   2;\n  2:   4,    2;\n  3:   8,    4;\n  4:  16,   16,   4;\n  5:  32,   32,   8;\n  6:  64,   96,  48,   8;\n  7: 128,  192,  96,  16;\n  8: 256,  512, 384, 128, 16;\n  9: 512, 1024, 768, 256, 32;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(n-1)*x)*P(n-1,x) - (4*x+2*x^2)*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[(2 + (n - 1)*x)*P[n - 1, x] - (4*x + 2*x^2)*D[P[n - 1, x], x]];\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] = (2 + (n - 1)*x)*v[i - 1] - (4*x + 2*x^2)*deriv(v[i - 1], x)); 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":["x \u21a6 x+1"],"edges":["e-5428ec8f88838d6b"],"text":"A274496 \u2192 x \u21a6 x+1"},"headline":"A274496 \u2192 x \u21a6 x+1","chain":null}
