{"id":"n-9a6ac7c630927289","revision":0,"first_index":1,"fingerprint":"f6-63795c237253f29b","name":"shift_x_plus_one of veronese_section_2_0 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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":[["1"],["1"],["3","1"],["8","5"],["22","18","1"],["60","61","9"],["164","199","49","1"],["448","632","218","13"],["1224","1968","873","96","1"],["3344","6036","3273","543","17"],["9136","18292","11723","2636","159","1"],["24960","54896","40596","11595","1100","21"],["68192","163424","136970","47604","6336","238","1"],["186304","483216","452648","185713","32282","1953","25"],["508992","1420528","1470744","696519","150666","13077","333","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","4","13","41","130","413","1311","4162","13213","41947","133168","422765","1342141","4260860"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1"],["3","1"],["8","5"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173301,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-5afebdeec36bb70a"]},{"id":173302,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-5afebdeec36bb70a"]},{"id":173303,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-5afebdeec36bb70a"]}],"region_exact":{},"incoming_edges":[{"id":"e-5afebdeec36bb70a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-9a6ac7c630927289","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":"6a7ad363bca0a83d8b0d4f86b7429225d0486fd4dd91f18bf30375dcc5f61d06","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-af589024565f1365","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    1;\n   3:    3,    1;\n   4:    8,    5;\n   5:   22,   18,    1;\n   6:   60,   61,    9;\n   7:  164,  199,   49,   1;\n   8:  448,  632,  218,  13;\n   9: 1224, 1968,  873,  96,  1;\n  10: 3344, 6036, 3273, 543, 17;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-4,x) + 2*x*P(n-3,x) + (2+x)*P(n-2,x) + 2*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 3 + x, P(4,x) = 8 + 5*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 3 + x;\nP[4, x_] := 8 + 5*x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 4, x] + 2*x*P[n - 3, x] + (2 + x)*P[n - 2, x] + 2*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 3 + x; v[4] = 8 + 5*x; for(n = 5, N, my(i = n); v[i] = x*v[i - 4] + 2*x*v[i - 3] + (2 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-918cdc60ffda35ef","e-5afebdeec36bb70a"],"text":"A063967 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A063967 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}