{"id":"n-9834c01ec55549ba","revision":0,"first_index":0,"fingerprint":"f6-6f025d7ab02f4150","name":"one_plus_derivative of shift_x_plus_one of Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["4","1"],["7","2"],["18","9","1"],["37","22","3"],["84","65","15","1"],["179","162","46","4"],["390","417","153","22","1"],["833","1014","441","80","5"],["1784","2457","1263","295","30","1"],["3791","5810","3420","956","125","6"],["8042","13609","9081","3004","505","39","1"],["16989","31446","23423","8912","1806","182","7"],["35804","72017","59343","25645","6160","798","49","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","5","9","28","62","165","391","983","2373","5830","14108","34281","82765","199817"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["-1","7","4"],["6","4","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["-24","-56","-38","-8"],["-2","35","34","8"],["27","30","8","0"],["5","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-252","-508","-383","-128","-16"],["-144","-208","-100","-16","0"],["-18","-17","-4","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172083,"position":0,"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-2fcdc6810470d0b9"]},{"id":172084,"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-2fcdc6810470d0b9"]},{"id":172085,"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-2fcdc6810470d0b9"]}],"region_exact":{},"incoming_edges":[{"id":"e-2fcdc6810470d0b9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-9834c01ec55549ba","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":"5fac77cc94e5312cfc7233599017b717cf418891552adef2a2c9fcb0ee578e14","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-fe5f0667f4ce56c9","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 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-c5ebb0acb0e1a953","kind":"fingerprint_related","node_a_id":"n-f2b038a33436dda1","node_b_id":"n-9834c01ec55549ba","fingerprint":"f6-6f025d7ab02f4150","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6f025d7ab02f4150"},"other_id":"n-f2b038a33436dda1","other":{"id":"n-f2b038a33436dda1","name":"one_plus_derivative of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","first_index":0,"fingerprint":"f6-6f025d7ab02f4150","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","1"],["7","2"]],"card_rows":[["1"],["4","1"],["7","2"],["18","9","1"]],"row_sums_preview":["1","5","9","28","62","165","391","983"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ac04e52510a48bb5","steps":["D+1"],"edges":["e-14d41bb220ff7187"],"text":"A249139 \u2192 D+1"},"headline":"A249139 \u2192 D+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:   1;\n  1:   1;\n  2:   4,    1;\n  3:   7,    2;\n  4:  18,    9,   1;\n  5:  37,   22,   3;\n  6:  84,   65,  15,  1;\n  7: 179,  162,  46,  4;\n  8: 390,  417, 153, 22, 1;\n  9: 833, 1014, 441, 80, 5;\n","recurrence_sentence":"The row polynomials satisfy (n^3+6*n^2-n+(4*n^2+7*n)*x+4*n*x^2)*P(n,x) = (5*n^3+27*n^2-2*n-24+(2*n^3+30*n^2+35*n-56)*x+(8*n^2+34*n-38)*x^2+(8*n-8)*x^3)*P(n-1,x) - (18*n^2+144*n+252+(17*n^2+208*n+508)*x+(4*n^2+100*n+383)*x^2+(16*n+128)*x^3+16*x^4)*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[((5*n^3 + 27*n^2 - 2*n - 24 + (2*n^3 + 30*n^2 + 35*n - 56)*x + (8*n^2 + 34*n - 38)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - (18*n^2 + 144*n + 252 + (17*n^2 + 208*n + 508)*x + (4*n^2 + 100*n + 383)*x^2 + (16*n + 128)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 + 6*n^2 - n + (4*n^2 + 7*n)*x + 4*n*x^2)]];\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] = ((5*n^3 + 27*n^2 - 2*n - 24 + (2*n^3 + 30*n^2 + 35*n - 56)*x + (8*n^2 + 34*n - 38)*x^2 + (8*n - 8)*x^3)*v[i - 1] - (18*n^2 + 144*n + 252 + (17*n^2 + 208*n + 508)*x + (4*n^2 + 100*n + 383)*x^2 + (16*n + 128)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 + 6*n^2 - n + (4*n^2 + 7*n)*x + 4*n*x^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-7d332fcb7bf9a0f7","steps":["basis A007318","D+1"],"edges":["e-10b46a3f983f65c9","e-2fcdc6810470d0b9"],"text":"A128100 \u2192 basis A007318 \u2192 D+1"},"headline":"A128100 \u2192 basis A007318 \u2192 D+1","chain":null}