{"id":"n-4021924eb530c03c","revision":0,"first_index":0,"fingerprint":"f6-ad8665a1ff262206","name":"shift_x_plus_one of derivative of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["2"],["4"],["14","8"],["32","24"],["82","96","24"],["188","272","96"],["438","792","432","64"],["984","2088","1440","320"],["2202","5424","4680","1600","160"],["4852","13536","13824","6080","960"],["10622","33192","39456","21760","5280","384"],["23056","79672","107712","70784","22400","2688"],["49762","188544","286200","219520","87360","16128","896"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","2","4","22","56","202","556","1726","4832","14066","39252","110694","306312","848410"],"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"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-13","-12"],["5","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-9","-17","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172079,"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-f012576891a3e620"]},{"id":172080,"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-6d6c1b443c41d0da","e-f012576891a3e620"]},{"id":172081,"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-6d6c1b443c41d0da","e-f012576891a3e620"]},{"id":172082,"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-6d6c1b443c41d0da"]}],"region_exact":{},"incoming_edges":[{"id":"e-6d6c1b443c41d0da","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-4021924eb530c03c","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"d58f60eafebd822698dbd60f3a185d33976795efc736fc019b3c02419de4f77b","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-0d89b6177f6a1dbb","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","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-f012576891a3e620","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-4021924eb530c03c","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":"5d6492d83cb9f3aa68042ae33ab56b84d5d3f929bd6cd956f5e4d9ef9eaee521","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-f8f37b87137bc296","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","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-e0353f0e5f6c11f0","kind":"fingerprint_related","node_a_id":"n-4021924eb530c03c","node_b_id":"n-573c4ed7a2758dc8","fingerprint":"f6-ad8665a1ff262206","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ad8665a1ff262206"},"other_id":"n-573c4ed7a2758dc8","other":{"id":"n-573c4ed7a2758dc8","name":"mobius_x_over_x_plus_one of theta_plus_one of A Jacobsthal triangle.","first_index":0,"fingerprint":"f6-ad8665a1ff262206","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","2"],["0","4","7"]],"card_rows":[["1"],["0","2"],["0","4","7"],["0","0","12","16"]],"row_sums_preview":["1","2","11","28","101","278","863","2416"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba196dcd01800741","steps":["\u03b8+1","x \u21a6 x/(x+1)"],"edges":["e-ecb19626d7d3a316","e-2ad9825d2a2c8673"],"text":"A113953 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A113953 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)","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:   0;\n  1:   0;\n  2:   2;\n  3:   4;\n  4:  14,    8;\n  5:  32,   24;\n  6:  82,   96,   24;\n  7: 188,  272,   96;\n  8: 438,  792,  432,  64;\n  9: 984, 2088, 1440, 320;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (5*n-13+(4*n-12)*x)*P(n-1,x) - (9+17*x+8*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n - 13 + (4*n - 12)*x)*P[n - 1, x] - (9 + 17*x + 8*x^2)*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((5*n - 13 + (4*n - 12)*x)*v[i - 1] - (9 + 17*x + 8*x^2)*deriv(v[i - 1], x))/(n - 2)); 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-e3d901814302c059","steps":["x \u21a6 x+1","D"],"edges":["e-31754865d296b758","e-6d6c1b443c41d0da"],"text":"A128099 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A128099 \u2192 x \u21a6 x+1 \u2192 D","chain":null}
