{"id":"n-d5c171da560bf838","revision":0,"first_index":1,"fingerprint":"f6-6a949130fb249910","name":"shift_x_plus_one of basis_map_A008277 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","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"],["1"],["3","1"],["5","2"],["12","8","1"],["24","19","3"],["56","57","15","1"],["123","144","46","4"],["292","401","166","24","1"],["682","1048","506","90","5"],["1667","2875","1643","380","35","1"],["4079","7713","4985","1356","155","6"],["10288","21291","15572","5039","750","48","1"],["26159","58479","47293","17424","3066","245","7"],["68026","163784","145989","61191","12845","1337","63","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","4","7","21","46","129","317","884","2331","6601","18294","52989","152673","453236"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172224,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-c74f23450b6a89cd"]},{"id":172225,"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-c74f23450b6a89cd"]},{"id":172226,"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-c74f23450b6a89cd"]}],"region_exact":{},"incoming_edges":[{"id":"e-c74f23450b6a89cd","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-d5c171da560bf838","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":"bfa8f2d93b16d36f5a6a4d6b76ae48bcc4cab46106b6b3a893be1a1c0ff4ff50","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-a4df05e4bf9e978a","role":"sequence","name":"basis_map_A008277 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward 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-f05a9e6ad3e1056a","kind":"fingerprint_related","node_a_id":"n-d5c171da560bf838","node_b_id":"n-ab47d782fbd81d4b","fingerprint":"f6-6a949130fb249910","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6a949130fb249910"},"other_id":"n-ab47d782fbd81d4b","other":{"id":"n-ab47d782fbd81d4b","name":"shift_x_plus_one of basis_map_A008277 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":"f6-6a949130fb249910","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["3","1"]],"card_rows":[["1"],["1"],["3","1"],["5","2"]],"row_sums_preview":["1","1","4","7","21","46","129","317"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75ccd0453a9b6cec","steps":["basis A008277","x \u21a6 x+1"],"edges":["e-67e9a817b66ff47d","e-694d7d59b95131b3"],"text":"A011973 \u2192 basis A008277 \u2192 x \u21a6 x+1"},"headline":"A011973 \u2192 basis A008277 \u2192 x \u21a6 x+1","chain":null}}],"oeis":[],"bfiles":[],"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:   1;\n   2:   1;\n   3:   3,    1;\n   4:   5,    2;\n   5:  12,    8,   1;\n   6:  24,   19,   3;\n   7:  56,   57,  15,  1;\n   8: 123,  144,  46,  4;\n   9: 292,  401, 166, 24, 1;\n  10: 682, 1048, 506, 90, 5;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-2,x) + (1+x)*d/dx P(n-2,x) + P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 2, x] + (1 + x)*D[P[n - 2, x], x] + P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (2 + x)*v[i - 2] + (1 + x)*deriv(v[i - 2], x) + v[i - 1]); 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-c8b3f39d03de798f","steps":["basis A008277","x \u21a6 x+1"],"edges":["e-658de443ccafc3cd","e-c74f23450b6a89cd"],"text":"A052553 \u2192 basis A008277 \u2192 x \u21a6 x+1"},"headline":"A052553 \u2192 basis A008277 \u2192 x \u21a6 x+1","chain":null}
