{"id":"n-a7581c3640cec184","revision":0,"first_index":0,"fingerprint":"f6-7b01bf956bad76c1","name":"shift_x_plus_one of veronese_section_2_0 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1"],["2","1"],["4","3"],["7","6"],["11","10"],["17","17","1"],["27","31","5"],["44","58","15"],["72","106","35"],["117","188","73","1"],["189","328","147","7"],["305","570","294","28"],["493","990","582","84"]],"degrees":[0,0,0,1,1,1,1,2,2,2,2,3,3,3,3],"row_sums":["1","1","1","3","7","13","21","35","63","117","213","379","671","1197","2149"],"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"],["1"],["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171315,"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-b97fbdeead455cda"]},{"id":171316,"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-b97fbdeead455cda"]},{"id":171317,"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-b97fbdeead455cda"]}],"region_exact":{},"incoming_edges":[{"id":"e-b97fbdeead455cda","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-a7581c3640cec184","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":"4671d7756cc3fe463ba6ff17f40e2818133991956ea9431e0c9c51aef4fa40e4","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-a2c58ea90c4c1ddc","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","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-047963f98d882114","kind":"fingerprint_related","node_a_id":"n-4a08d50eedd2c85f","node_b_id":"n-a7581c3640cec184","fingerprint":"f6-7b01bf956bad76c1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7b01bf956bad76c1"},"other_id":"n-4a08d50eedd2c85f","other":{"id":"n-4a08d50eedd2c85f","name":"shift_x_plus_one of veronese_section_2_0 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","first_index":1,"fingerprint":"f6-7b01bf956bad76c1","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1"]],"card_rows":[["1"],["1"],["1"],["1"]],"row_sums_preview":["1","1","1","1","3","7","13","21"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8b3f39d03de798f","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-86342ef0069ad106","e-f66b3841295af911"],"text":"A052553 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A052553 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-d838c68eb295cadd","kind":"fingerprint_related","node_a_id":"n-87700d0fc3ae2b6a","node_b_id":"n-a7581c3640cec184","fingerprint":"f6-7b01bf956bad76c1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7b01bf956bad76c1"},"other_id":"n-87700d0fc3ae2b6a","other":{"id":"n-87700d0fc3ae2b6a","name":"shift_x_plus_one of veronese_section_2_0 of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","first_index":0,"fingerprint":"f6-7b01bf956bad76c1","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1"]],"card_rows":[["1"],["1"],["1"],["1"]],"row_sums_preview":["1","1","1","1","3","7","13","21"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75ccd0453a9b6cec","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-054f5d56a30dc717","e-ec6558d6a87628bc"],"text":"A011973 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A011973 \u2192 Veronese 2 \u2192 x \u21a6 x+1","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:  1;\n  1:  1;\n  2:  1;\n  3:  2,  1;\n  4:  4,  3;\n  5:  7,  6;\n  6: 11, 10;\n  7: 17, 17,  1;\n  8: 27, 31,  5;\n  9: 44, 58, 15;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-4,x) - P(n-2,x) + 2*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1, P(3,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 4, x] - P[n - 2, x] + 2*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1; v[4] = 2 + x; for(n = 4, N, my(i = n + 1); v[i] = (1 + x)*v[i - 4] - v[i - 2] + 2*v[i - 1]); 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-569a801b809b6abd","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-fc2a267a1cf18403","e-b97fbdeead455cda"],"text":"A169803 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A169803 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}