{"id":"n-bb656bb28f34987e","revision":0,"first_index":0,"fingerprint":"f6-7e54cdbe5f0f29a0","name":"shift_x_plus_one of basis_map_A013609 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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"],["5","2"],["28","22","4"],["155","182","68","8"],["859","1342","760","184","16"],["4760","9284","7052","2600","464","32"],["26377","61684","58792","29176","7936","1120","64"],["146165","398546","457052","285368","104624","22496","2624","128"],["809956","2522866","3382096","2546056","1176016","340960","60544","6016","256"],["4488275","15722210","24124460","21264680","11856800","4333568","1037504","156800","13568","512"],["24871243","96776110","167258740","168974680","110433520","48756352","14718208","2998144","394240","30208","1024"],["137821040","589766216","1133663720","1291433840","968216720","501363104","182883392","46972544","8321792","968192","66560","2048"],["763718929","3564543976","7543179472","9565938160","8093201200","4810385056","2058810496","639060352","142733056","22363648","2332672","145408","4096"],["4232057765","21395098466","49425508904","69057678704","65085400160","43678850432","21466198976","7819607168","2110696448","416832512","58526720","5531648","315392","8192"],["23451445612","127660677646","319676367820","487923584752","506953838128","379182610048","210485897344","88065235840","27899016448","6658112512","1178023936","149813248","12943360","679936","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","54","413","3161","24192","185149","1417003","10844766","82998377","635212469","4861479168","37206416521","284752311487","2179298263014"],"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":[["40","16"],["-32","-8"],["12","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["200","176","32"],["-74","-56","-8"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["111","158","68","8"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179115,"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-1ba3c61a745c02de"]},{"id":179116,"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-1ba3c61a745c02de"]},{"id":179117,"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-1ba3c61a745c02de"]}],"region_exact":{},"incoming_edges":[{"id":"e-1ba3c61a745c02de","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-bb656bb28f34987e","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":"967005fc7cdb73c84f71061a01c6613bca159e9378667561066d4a2557d445dc","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-da6c9018f05787b2","role":"sequence","name":"basis_map_A013609 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*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-841dacf256277971","kind":"candidate_equal","node_a_id":"n-bb656bb28f34987e","node_b_id":"n-1af6da463398282c","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-1af6da463398282c","other":{"id":"n-1af6da463398282c","name":"basis_map_A007318 of basis_map_A013609 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","first_index":0,"fingerprint":"f6-7e54cdbe5f0f29a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","2"],["28","22","4"]],"card_rows":[["1"],["5","2"],["28","22","4"],["155","182","68","8"]],"row_sums_preview":["1","7","54","413","3161","24192","185149","1417003"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d5c14b9fec875ecc","steps":["basis A013609","basis A007318"],"edges":["e-eac85dfbd4ac8581","e-5a02e3d857e93d03"],"text":"A121574 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A121574 \u2192 basis A013609 \u2192 basis A007318","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:       1;\n  1:       5,        2;\n  2:      28,       22,        4;\n  3:     155,      182,       68,        8;\n  4:     859,     1342,      760,      184,       16;\n  5:    4760,     9284,     7052,     2600,      464,      32;\n  6:   26377,    61684,    58792,    29176,     7936,    1120,      64;\n  7:  146165,   398546,   457052,   285368,   104624,   22496,    2624,    128;\n  8:  809956,  2522866,  3382096,  2546056,  1176016,  340960,   60544,   6016,   256;\n  9: 4488275, 15722210, 24124460, 21264680, 11856800, 4333568, 1037504, 156800, 13568, 512;\n","recurrence_sentence":"The row polynomials satisfy 4*n*P(n,x) = (12*n^2-32*n+40-(8*n-16)*x)*P(n-1,x) - (74*n-200+(56*n-176)*x+(8*n-32)*x^2)*d/dx P(n-1,x) + (111+158*x+68*x^2+8*x^3)*d^2/dx^2 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[((12*n^2 - 32*n + 40 - (8*n - 16)*x)*P[n - 1, x] - (74*n - 200 + (56*n - 176)*x + (8*n - 32)*x^2)*D[P[n - 1, x], x] + (111 + 158*x + 68*x^2 + 8*x^3)*D[P[n - 1, x], {x, 2}])/(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] = ((12*n^2 - 32*n + 40 - (8*n - 16)*x)*v[i - 1] - (74*n - 200 + (56*n - 176)*x + (8*n - 32)*x^2)*deriv(v[i - 1], x) + (111 + 158*x + 68*x^2 + 8*x^3)*deriv(deriv(v[i - 1], x), 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-d5c14b9fec875ecc","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-eac85dfbd4ac8581","e-1ba3c61a745c02de"],"text":"A121574 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A121574 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}