{"id":"n-4017c63c30535842","revision":0,"first_index":0,"fingerprint":"f6-fad358f622a81bb9","name":"shift_x_plus_one of veronese_section_2_1 of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","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"],["3"],["6","1"],["17","10"],["45","37","1"],["116","126","21"],["305","420","129","1"],["799","1318","570","36"],["2090","4023","2283","334","1"],["5473","12054","8481","1936","55"],["14329","35515","29793","9305","720","1"],["37512","103340","100782","40248","5395","78"],["98209","297704","331038","161298","31150","1371","1"],["257115","850460","1062092","611652","155833","13006","105"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","3","7","27","83","263","855","2723","8731","27999","89663","287355","920771","2950263"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["3"],["6","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-2","-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172770,"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-9ee38dd60d247cc4"]},{"id":172771,"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-9ee38dd60d247cc4"]},{"id":172772,"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-9ee38dd60d247cc4"]}],"region_exact":{},"incoming_edges":[{"id":"e-9ee38dd60d247cc4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-4017c63c30535842","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":"0f921ed0444e175e84e26378c2fb8cb28154e9a5d850cacf9c7872adfdbb9e63","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-3d82b1c51e4bade1","role":"sequence","name":"veronese_section_2_1 of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","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:    0;\n  1:    1;\n  2:    3;\n  3:    6,    1;\n  4:   17,   10;\n  5:   45,   37,    1;\n  6:  116,  126,   21;\n  7:  305,  420,  129,   1;\n  8:  799, 1318,  570,  36;\n  9: 2090, 4023, 2283, 334, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+2*x+x^2)*P(n-4,x) + (2+2*x)*P(n-3,x) + (1+2*x)*P(n-2,x) + 2*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 3, P(3,x) = 6 + x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 3;\nP[3, x_] := 6 + x;\nP[n_, x_] := P[n, x] = Expand[-(1 + 2*x + x^2)*P[n - 4, x] + (2 + 2*x)*P[n - 3, x] + (1 + 2*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] = 0; v[2] = 1; v[3] = 3; v[4] = 6 + x; for(n = 4, N, my(i = n + 1); v[i] = -(1 + 2*x + x^2)*v[i - 4] + (2 + 2*x)*v[i - 3] + (1 + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6758fe7729845be8","steps":["Veronese 2 (j=1)","x \u21a6 x+1"],"edges":["e-33a38ab7b1aab5c6","e-9ee38dd60d247cc4"],"text":"A054142 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1"},"headline":"A054142 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1","chain":null}