{"id":"n-9aa9e3dd2403bca2","revision":1,"first_index":0,"fingerprint":"f6-0dd3ed80d2a0b395","name":"shift_x_plus_one of Riordan array ((1+x)^3,x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["8","12","6","1"],["8","20","18","7","1"],["8","28","38","25","8","1"],["8","36","66","63","33","9","1"],["8","44","102","129","96","42","10","1"],["8","52","146","231","225","138","52","11","1"],["8","60","198","377","456","363","190","63","12","1"],["8","68","258","575","833","819","553","253","75","13","1"],["8","76","326","833","1408","1652","1372","806","328","88","14","1"],["8","84","402","1159","2241","3060","3024","2178","1134","416","102","15","1"],["8","92","486","1561","3400","5301","6084","5202","3312","1550","518","117","16","1"],["8","100","578","2047","4961","8701","11385","11286","8514","4862","2068","635","133","17","1"],["8","108","678","2625","7008","13662","20086","22671","19800","13376","6930","2703","768","150","18","1"],["8","116","786","3303","9633","20670","33748","42757","42471","33176","20306","9633","3471","918","168","19","1"],["8","124","902","4089","12936","30303","54418","76505","85228","75647","53482","29939","13104","4389","1086","187","20","1"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["27","54","108","216","432","864","1728","3456","6912","13824","27648","55296","110592","221184","442368"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["8","12","6","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160694,"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-56c62e2300be82c0"]},{"id":160695,"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-09b769bb10417be2","e-56c62e2300be82c0"]},{"id":160696,"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-09b769bb10417be2","e-56c62e2300be82c0"]},{"id":160697,"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-09b769bb10417be2"]},{"id":160698,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-0331d381cbbd8a41"]}],"region_exact":{},"incoming_edges":[{"id":"e-09b769bb10417be2","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-9aa9e3dd2403bca2","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":"69a13680c3ae08a3d12c28e16d7100ef070a29b5f65300933b0ca1e2830e1924","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e1d160f3b707da31","role":"sequence","name":"Riordan array ((1+x)^3,x).","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-0331d381cbbd8a41","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-9aa9e3dd2403bca2","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"06506dfbbc1e74190f94fb69b9ceae73d516d0dc408d86f8aa7d54df290251dc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-f48bed27dbc29c5f","role":"sequence","name":"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":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-f48bed27dbc29c5f","role":"sequence","name":"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":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]},{"id":"e-56c62e2300be82c0","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-9aa9e3dd2403bca2","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":"cb238558c6416fc2fb0a4e99156feee79a74e332187423b7f3003cf9944b5678","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-e1d160f3b707da31","role":"sequence","name":"Riordan array ((1+x)^3,x).","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-a8ab6569f5bcfbfb","kind":"oeis_match","node_a_id":"n-9aa9e3dd2403bca2","node_b_id":null,"fingerprint":null,"oeis_id":"A175806","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A175806","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A175806","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-9aa9e3dd2403bca2","oeis_id":"A175806","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":31,"oeis_terms_available":31,"term_count":200,"raw_size":7267}],"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: 8, 12,   6,    1;\n  1: 8, 20,  18,    7,    1;\n  2: 8, 28,  38,   25,    8,    1;\n  3: 8, 36,  66,   63,   33,    9,    1;\n  4: 8, 44, 102,  129,   96,   42,   10,    1;\n  5: 8, 52, 146,  231,  225,  138,   52,   11,    1;\n  6: 8, 60, 198,  377,  456,  363,  190,   63,   12,   1;\n  7: 8, 68, 258,  575,  833,  819,  553,  253,   75,  13,   1;\n  8: 8, 76, 326,  833, 1408, 1652, 1372,  806,  328,  88,  14,  1;\n  9: 8, 84, 402, 1159, 2241, 3060, 3024, 2178, 1134, 416, 102, 15, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 12*x + 6*x^2 + x^3.","mathematica":"P[0, x_] := 8 + 12*x + 6*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[(1 + 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] = 8 + 12*x + 6*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (1 + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f48bed27dbc29c5f","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-0331d381cbbd8a41"],"text":"A391533 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A391533 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}