{"id":"n-4f19d6ed4b208f66","revision":0,"first_index":0,"fingerprint":"f6-b4388d7923690e8a","name":"derivative 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":-4,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],[],[],["8"],["24"],["48","48"],["80","192"],["120","480","192"],["168","960","960"],["224","1680","2880","640"],["288","2688","6720","3840"],["360","4032","13440","13440","1920"],["440","5760","24192","35840","13440"],["528","7920","40320","80640","53760","5376"]],"degrees":[-1,-1,-1,-1,0,0,1,1,2,2,3,3,4,4,5],"row_sums":["0","0","0","0","8","24","96","272","792","2088","5424","13536","33192","79672","188544"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["0"],["0"],["8"]],"recurrence":{"denominator":{"coeffs":[["-4"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-20"],["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172067,"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-841e59d68ceccf63"]},{"id":172068,"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-841e59d68ceccf63"]},{"id":172069,"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-841e59d68ceccf63"]}],"region_exact":{},"incoming_edges":[{"id":"e-841e59d68ceccf63","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-4f19d6ed4b208f66","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":"d7f2cab6e30ea6bef78d41bdf00f4e2af584ecd05afe0c54b69b79e01a9299f8","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-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":[],"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:   0;\n  3:   0;\n  4:   8;\n  5:  24;\n  6:  48,  48;\n  7:  80, 192;\n  8: 120, 480, 192;\n  9: 168, 960, 960;\n","recurrence_sentence":"The row polynomials satisfy (n-4)*P(n,x) = (n-2+(4*n-20)*x)*P(n-1,x) - (x+8*x^2)*d/dx P(n-1,x) for n >= 5, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 0, P(3,x) = 0, P(4,x) = 8.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 0;\nP[4, x_] := 8;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (4*n - 20)*x)*P[n - 1, x] - (x + 8*x^2)*D[P[n - 1, x], x])/(n - 4)]];\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] = 0; v[4] = 0; v[5] = 8; for(n = 5, N, my(i = n + 1); v[i] = ((n - 2 + (4*n - 20)*x)*v[i - 1] - (x + 8*x^2)*deriv(v[i - 1], x))/(n - 4)); 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":["D","D"],"edges":["e-3a54b8685030d0ee","e-841e59d68ceccf63"],"text":"A128099 \u2192 D \u2192 D"},"headline":"A128099 \u2192 D \u2192 D","chain":null}