{"id":"n-d653bfeaea2b665f","revision":0,"first_index":0,"fingerprint":"f6-03f9938138a9e92b","name":"shift_x_plus_one of basis_map_A013609 of Even row Pascal-square read by antidiagonals.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["7","4"],["40","44","12"],["217","348","184","32"],["1159","2428","1884","640","80"],["6160","15896","16200","8128","2000","192"],["32689","100152","126256","83616","30576","5824","448"],["173383","614836","923208","759200","368160","104832","16128","1024"],["919480","3703748","6452916","6339008","3830320","1452672","336000","43008","2304"],["4875913","21990036","43604968","49814080","36051600","17091776","5286912","1022976","111360","5120"],["25856071","129066412","286995780","373817376","315264880","179454912","69598144","18078720","2990592","281600","11264"],["137109280","750467888","1849407312","2705158272","2605470432","1731542400","808116736","263926784","58823424","8459264","698368","24576"],["727060321","4329841776","11712539488","19009165760","20587750560","15649136256","8542084992","3364248576","945218560","183772160","23282688","1703936","53248"],["3855438727","24817611364","73108425936","130373134528","156842154560","134224796160","83917698816","38719168512","13144525056","3229557760","554977280","62619648","4100096","114688"],["20444528200","141451734620","450750229020","876093312640","1159137717920","1103010295296","777303674112","411325172736","163842059520","48731438080","10607387648","1628430336","165122048","9748480","245760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","96","781","6191","48576","379561","2960771","23079456","179854741","1401415751","10919204736","85075858321","662854323131","5164501096416"],"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":[["-12","-8"],["40","24"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-39","-50","-16"]]},"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-80e18b2692261096","recurrence":{"first_index":0,"initial_polynomials":[["1"],["7","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-9","-12","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["7","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180080,"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-53fc0a74c67e1a12"]},{"id":180081,"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-293017fb435bbee2","e-53fc0a74c67e1a12"]},{"id":180082,"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-293017fb435bbee2","e-53fc0a74c67e1a12"]},{"id":180083,"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-293017fb435bbee2"]},{"id":180084,"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-293017fb435bbee2"]}],"region_exact":{},"incoming_edges":[{"id":"e-293017fb435bbee2","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-d653bfeaea2b665f","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":"8bd5d2252d747d1bbb61939f02ca3ac4430423b3ec0090495297e2a3f3fe7ad9","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-7c7a913d07444057","role":"sequence","name":"basis_map_A013609 of Even row Pascal-square read by antidiagonals.","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-80e18b2692261096","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-d653bfeaea2b665f","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":"479ed42f2c4adf0673aad7ef81e2cb6c693df37c1c7ae1f89f9babd18639021a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-53fc0a74c67e1a12","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-d653bfeaea2b665f","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":"52ef5c4d3b678c7618cc22f929d5fd8f381963ecd7b24fb321f00cf3bdbb2617","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-7c7a913d07444057","role":"sequence","name":"basis_map_A013609 of Even row Pascal-square read by antidiagonals.","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":[],"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:       7,        4;\n  2:      40,       44,       12;\n  3:     217,      348,      184,       32;\n  4:    1159,     2428,     1884,      640,       80;\n  5:    6160,    15896,    16200,     8128,     2000,      192;\n  6:   32689,   100152,   126256,    83616,    30576,     5824,     448;\n  7:  173383,   614836,   923208,   759200,   368160,   104832,   16128,    1024;\n  8:  919480,  3703748,  6452916,  6339008,  3830320,  1452672,  336000,   43008,   2304;\n  9: 4875913, 21990036, 43604968, 49814080, 36051600, 17091776, 5286912, 1022976, 111360, 5120;\n","recurrence_sentence":"The row polynomials satisfy 4*n*P(n,x) = (40*n-12+(24*n-8)*x)*P(n-1,x) - (39+50*x+16*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[((40*n - 12 + (24*n - 8)*x)*P[n - 1, x] - (39 + 50*x + 16*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] = ((40*n - 12 + (24*n - 8)*x)*v[i - 1] - (39 + 50*x + 16*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-79ebafb266da3b41","steps":["basis A013609","basis A007318"],"edges":["e-1a56a7670c2d3c41","e-293017fb435bbee2"],"text":"A172431 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A172431 \u2192 basis A013609 \u2192 basis A007318","chain":null}