{"id":"n-54d7b058e1600620","revision":0,"first_index":0,"fingerprint":"f6-afc8bef80990b82e","name":"theta_plus_one_squared of Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k different colors.","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"],["1","4"],["1","8"],["1","16","27"],["1","24","54"],["1","40","189","192"],["1","56","324","384"],["1","88","837","2112","1500"],["1","120","1350","3840","3000"],["1","184","3105","14400","24000","12960"],["1","248","4860","24960","45000","25920"],["1","376","10557","79680","232500","285120","123480"],["1","504","16254","134400","420000","544320","246960"],["1","760","34209","393792","1785000","3719520","3580920","1290240"],["1","1016","52164","653184","3150000","6894720","6914880","2580480"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["1","5","9","44","79","422","765","4538","8311","54650","100989","731714","1362439","10804442","20246445"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","4"],["1","8"],["1","16","27"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-8","22"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","-2","0","26"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","4","5"]]},"deriv_order":2,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2","-4"]]},"deriv_order":1,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172335,"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-2f78ee142f184ee3"]},{"id":172336,"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-2f78ee142f184ee3"]},{"id":172337,"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-2f78ee142f184ee3"]}],"region_exact":{},"incoming_edges":[{"id":"e-2f78ee142f184ee3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-54d7b058e1600620","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"e34729f4b5503dca136927ca7e30cafee105ff45d54f6fa5dc4257300ebf5541","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-f1291d4c6c67167e","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k different colors.","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: 1;\n  1: 1,   4;\n  2: 1,   8;\n  3: 1,  16,   27;\n  4: 1,  24,   54;\n  5: 1,  40,  189,   192;\n  6: 1,  56,  324,   384;\n  7: 1,  88,  837,  2112,  1500;\n  8: 1, 120, 1350,  3840,  3000;\n  9: 1, 184, 3105, 14400, 24000, 12960;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(8*x-22*x^2)*P(n-4,x) - (2*x-26*x^3)*d/dx P(n-4,x) - (x^2-4*x^3-5*x^4)*d^2/dx^2 P(n-4,x) + (1+8*x)*P(n-2,x) + (2*x-4*x^2)*d/dx P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 8*x, P(3,x) = 1 + 16*x + 27*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[2, x_] := 1 + 8*x;\nP[3, x_] := 1 + 16*x + 27*x^2;\nP[n_, x_] := P[n, x] = Expand[-(8*x - 22*x^2)*P[n - 4, x] - (2*x - 26*x^3)*D[P[n - 4, x], x] - (x^2 - 4*x^3 - 5*x^4)*D[P[n - 4, x], {x, 2}] + (1 + 8*x)*P[n - 2, x] + (2*x - 4*x^2)*D[P[n - 2, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; v[3] = 1 + 8*x; v[4] = 1 + 16*x + 27*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(8*x - 22*x^2)*v[i - 4] - (2*x - 26*x^3)*deriv(v[i - 4], x) - (x^2 - 4*x^3 - 5*x^4)*deriv(deriv(v[i - 4], x), x) + (1 + 8*x)*v[i - 2] + (2*x - 4*x^2)*deriv(v[i - 2], x)); 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-f1291d4c6c67167e","steps":["(\u03b8+1)\u00b2"],"edges":["e-2f78ee142f184ee3"],"text":"A305540 \u2192 (\u03b8+1)\u00b2"},"headline":"A305540 \u2192 (\u03b8+1)\u00b2","chain":null}