{"id":"n-d945d6c9616de0d7","revision":0,"first_index":0,"fingerprint":"f6-7bd50a5129aedca7","name":"shift_x_plus_one of basis_map_A013611 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.","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":false,"rows":[["1"],["1"],["6","4"],["11","8"],["41","52","16"],["96","136","48"],["301","560","336","64"],["781","1624","1120","256"],["2286","5628","5040","1920","256"],["6191","16872","17136","7680","1280"],["17621","54156","64848","37440","10240","1024"],["48576","163280","218016","144384","47360","6144"],["136681","504544","758880","590976","248320","52224","4096"],["379561","1515248","2502080","2184960","1062656","272384","28672"],["1062966","4584692","8314656","8175360","4668160","1526784","258048","16384"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","10","19","109","280","1261","3781","15130","49159","185329","627760","2295721","7945561","28607050"],"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":{"denominator":{"coeffs":[["0"],["4"]]},"terms":[{"coeff":{"coeffs":[["-40","-32"],["44","32"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-105","-164","-64"]]},"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-ba56ecf6342e94db","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173201,"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-b1df3c46b185b4ef"]},{"id":173202,"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-802e10a370537b89","e-b1df3c46b185b4ef"]},{"id":173203,"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-802e10a370537b89","e-b1df3c46b185b4ef"]},{"id":173204,"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-802e10a370537b89"]},{"id":173205,"position":4,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-802e10a370537b89"]}],"region_exact":{},"incoming_edges":[{"id":"e-802e10a370537b89","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-d945d6c9616de0d7","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":"aada10e8494937556b230a3f59cc9f9a5f6b3726ecce0f50e0178c66de978b17","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e1fb974f437104f4","role":"sequence","name":"basis_map_A013611 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.","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-ba56ecf6342e94db","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-d945d6c9616de0d7","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":"e2347e56e824fe1684c630ca51a292a6a59008905445aaf054f802977e979abf","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-b1df3c46b185b4ef","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-d945d6c9616de0d7","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":"b8e92215cd7ef2dc86b9caacb5a9a2cdb98719ab9f7e0c4febcfb4652e8c0244","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-e1fb974f437104f4","role":"sequence","name":"basis_map_A013611 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.","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-18492ac4f304bdcb","kind":"fingerprint_related","node_a_id":"n-44655160843a1362","node_b_id":"n-d945d6c9616de0d7","fingerprint":"f6-7bd50a5129aedca7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7bd50a5129aedca7"},"other_id":"n-44655160843a1362","other":{"id":"n-44655160843a1362","name":"shift_x_plus_one of basis_map_A013611 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","first_index":1,"fingerprint":"f6-7bd50a5129aedca7","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["6","4"]],"card_rows":[["1"],["1"],["6","4"],["11","8"]],"row_sums_preview":["1","1","10","19","109","280","1261","3781"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8b3f39d03de798f","steps":["basis A013611","basis A007318"],"edges":["e-23dacb32496f8efb","e-d05886483ef03702"],"text":"A052553 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A052553 \u2192 basis A013611 \u2192 basis A007318","chain":null}},{"id":"l-668f4194c1dd7557","kind":"fingerprint_related","node_a_id":"n-d945d6c9616de0d7","node_b_id":"n-e88e218539d1c5b8","fingerprint":"f6-7bd50a5129aedca7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7bd50a5129aedca7"},"other_id":"n-e88e218539d1c5b8","other":{"id":"n-e88e218539d1c5b8","name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","first_index":0,"fingerprint":"f6-7bd50a5129aedca7","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["11","8"]],"card_rows":[["1"],["6","4"],["11","8"],["41","52","16"]],"row_sums_preview":["1","10","19","109","280","1261","3781","15130"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ac04e52510a48bb5","steps":["basis A013609","basis A013609"],"edges":["e-3f48f2d5eb407d05","e-4061325ce955aa51"],"text":"A249139 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A249139 \u2192 basis A013609 \u2192 basis A013609","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:    6,     4;\n  3:   11,     8;\n  4:   41,    52,    16;\n  5:   96,   136,    48;\n  6:  301,   560,   336,   64;\n  7:  781,  1624,  1120,  256;\n  8: 2286,  5628,  5040, 1920,  256;\n  9: 6191, 16872, 17136, 7680, 1280;\n","recurrence_sentence":"The row polynomials satisfy 4*n*P(n,x) = (44*n-40+(32*n-32)*x)*P(n-1,x) - (105+164*x+64*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[Together[((44*n - 40 + (32*n - 32)*x)*P[n - 1, x] - (105 + 164*x + 64*x^2)*D[P[n - 1, x], x])/(4*n)]];\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] = ((44*n - 40 + (32*n - 32)*x)*v[i - 1] - (105 + 164*x + 64*x^2)*deriv(v[i - 1], x))/(4*n)); 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-75ccd0453a9b6cec","steps":["basis A013611","basis A007318"],"edges":["e-2bd8266472adce93","e-802e10a370537b89"],"text":"A011973 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A011973 \u2192 basis A013611 \u2192 basis A007318","chain":null}