{"id":"n-3f8380ad8b34c71c","revision":0,"first_index":0,"fingerprint":"f6-457683072c1b3e64","name":"basis_map_A013609 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","8"],["11","16"],["41","104","64"],["96","272","192"],["301","1120","1344","512"],["781","3248","4480","2048"],["2286","11256","20160","15360","4096"],["6191","33744","68544","61440","20480"],["17621","108312","259392","299520","163840","32768"],["48576","326560","872064","1155072","757760","196608"],["136681","1009088","3035520","4727808","3973120","1671168","262144"],["379561","3030496","10008320","17479680","17002496","8716288","1835008"],["1062966","9169384","33258624","65402880","74690560","48857088","16515072","2097152"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","14","27","209","560","3277","10557","53158","190399","881453","3356640","14815529","58451849","251053726"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":198973,"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-a04f47a073bbf171"]},{"id":198974,"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-a04f47a073bbf171"]},{"id":198975,"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-a04f47a073bbf171"]},{"id":198976,"position":3,"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-a04f47a073bbf171"]}],"region_exact":{},"incoming_edges":[{"id":"e-a04f47a073bbf171","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-3f8380ad8b34c71c","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":"b215ee6657b23e80f7f9dbffd56ede2a40c946822b633ff4450037cde3ed896a","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]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-5fa6e82700abc4ce","kind":"fingerprint_related","node_a_id":"n-d059454219a91c67","node_b_id":"n-3f8380ad8b34c71c","fingerprint":"f6-457683072c1b3e64","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-457683072c1b3e64"},"other_id":"n-d059454219a91c67","other":{"id":"n-d059454219a91c67","name":"basis_map_A013609 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":"f6-457683072c1b3e64","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["6","8"]],"card_rows":[["1"],["1"],["6","8"],["11","16"]],"row_sums_preview":["1","1","14","27","209","560","3277","10557"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8b3f39d03de798f","steps":["basis A013611","basis A013609"],"edges":["e-23dacb32496f8efb","e-3ffceb8e5157e8a4"],"text":"A052553 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A052553 \u2192 basis A013611 \u2192 basis A013609","chain":null}},{"id":"l-d442d3b59156909b","kind":"fingerprint_related","node_a_id":"n-3f8380ad8b34c71c","node_b_id":"n-e2feae3cdfd3f927","fingerprint":"f6-457683072c1b3e64","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-457683072c1b3e64"},"other_id":"n-e2feae3cdfd3f927","other":{"id":"n-e2feae3cdfd3f927","name":"basis_map_A013609 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":"f6-457683072c1b3e64","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","8"],["11","16"]],"card_rows":[["1"],["6","8"],["11","16"],["41","104","64"]],"row_sums_preview":["1","14","27","209","560","3277","10557","53158"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-569a801b809b6abd","steps":["basis A013611","basis A013609"],"edges":["e-5fe58c53530d92d6","e-03e42ab978ab9814"],"text":"A169803 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A169803 \u2192 basis A013611 \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,     8;\n  3:   11,    16;\n  4:   41,   104,    64;\n  5:   96,   272,   192;\n  6:  301,  1120,  1344,   512;\n  7:  781,  3248,  4480,  2048;\n  8: 2286, 11256, 20160, 15360,  4096;\n  9: 6191, 33744, 68544, 61440, 20480;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+8*x)*P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(5 + 8*x)*P[n - 2, x] + 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; for(n = 2, N, my(i = n + 1); v[i] = (5 + 8*x)*v[i - 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-75ccd0453a9b6cec","steps":["basis A013611","basis A013609"],"edges":["e-2bd8266472adce93","e-a04f47a073bbf171"],"text":"A011973 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A011973 \u2192 basis A013611 \u2192 basis A013609","chain":null}