{"id":"n-67c59884b8050215","revision":0,"first_index":0,"fingerprint":"f6-57929f77c387d0af","name":"basis_map_A007318 of veronese_section_2_1 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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":[["4"],["9","1"],["18","6"],["36","21","1"],["72","60","8"],["144","156","37","1"],["288","384","134","10"],["576","912","424","57","1"],["1152","2112","1232","248","12"],["2304","4800","3376","920","81","1"],["4608","10752","8864","3072","410","14"],["9216","23808","22528","9520","1740","109","1"],["18432","52224","55808","27904","6552","628","16"],["36864","113664","135424","78336","22624","2996","141","1"],["73728","245760","323072","212480","73152","12544","910","18"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["4","10","24","58","140","338","816","1970","4756","11482","27720","66922","161564","390050","941664"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4"],["9","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195343,"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-c4a79027e85492f7"]},{"id":195344,"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-c4a79027e85492f7"]},{"id":195345,"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-c4a79027e85492f7"]},{"id":195346,"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-c4a79027e85492f7"]}],"region_exact":{},"incoming_edges":[{"id":"e-c4a79027e85492f7","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-67c59884b8050215","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":"a1869034a3150c72ef81d273a34cda1a7c60a43d7cea384020f1948e24a7089a","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-6feee566fc77ea06","role":"sequence","name":"veronese_section_2_1 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for 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-1d8f7bcc3275b02d","kind":"candidate_equal","node_a_id":"n-71b74177b3b9661f","node_b_id":"n-67c59884b8050215","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-71b74177b3b9661f","other":{"id":"n-71b74177b3b9661f","name":"shift_x_plus_one of veronese_section_2_1 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","first_index":0,"fingerprint":"f6-57929f77c387d0af","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["4"],["9","1"],["18","6"]],"card_rows":[["4"],["9","1"],["18","6"],["36","21","1"]],"row_sums_preview":["4","10","24","58","140","338","816","1970"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f48bed27dbc29c5f","steps":["Veronese 2 (j=1)","x \u21a6 x+1"],"edges":["e-a61d9f5d8a4277bd","e-e6a8292aab7e0c23"],"text":"A391533 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1"},"headline":"A391533 \u2192 Veronese 2 (j=1) \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:    4;\n  1:    9,    1;\n  2:   18,    6;\n  3:   36,   21,    1;\n  4:   72,   60,    8;\n  5:  144,  156,   37,   1;\n  6:  288,  384,  134,  10;\n  7:  576,  912,  424,  57,  1;\n  8: 1152, 2112, 1232, 248, 12;\n  9: 2304, 4800, 3376, 920, 81, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-2,x) + 2*P(n-1,x) for n >= 2, with P(0,x) = 4, P(1,x) = 9 + x.","mathematica":"P[0, x_] := 4;\nP[1, x_] := 9 + x;\nP[n_, x_] := P[n, x] = Expand[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] = 4; v[2] = 9 + x; for(n = 2, N, my(i = n + 1); v[i] = x*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-f48bed27dbc29c5f","steps":["Veronese 2 (j=1)","basis A007318"],"edges":["e-a61d9f5d8a4277bd","e-c4a79027e85492f7"],"text":"A391533 \u2192 Veronese 2 (j=1) \u2192 basis A007318"},"headline":"A391533 \u2192 Veronese 2 (j=1) \u2192 basis A007318","chain":null}
