{"id":"n-aea9a4160a820c9e","revision":0,"first_index":0,"fingerprint":"f6-acf6020f9423e8c7","name":"theta_plus_one_squared of basis_map_A008277 of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","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"],["2","4"],["3","8"],["5","24","9"],["8","52","27"],["13","120","108","16"],["21","256","306","64"],["34","548","900","320","25"],["55","1144","2394","1120","125"],["89","2376","6327","4000","750","36"],["144","4884","16083","12736","3125","216"],["233","9992","40410","39984","13125","1512","49"],["377","20336","99648","119520","48650","7308","343"],["610","41252","243360","351296","176750","35280","2744","64"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","6","11","38","87","257","647","1827","4838","13578","37188","105305","296182","851356"],"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"],["2","4"],["3","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-4","9"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","-2","4","4"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","4"]]},"deriv_order":2,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["-1","-4"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","-2","4"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2","-4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172252,"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-cbd7f1509507a974"]},{"id":172253,"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-cbd7f1509507a974"]},{"id":172254,"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-cbd7f1509507a974"]}],"region_exact":{},"incoming_edges":[{"id":"e-cbd7f1509507a974","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-aea9a4160a820c9e","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":"fbbf9db3646fa17c0ef83a7d501fad51e6487c298926e7762f05a11c5aad8a2c","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-c0b420d860aa68c5","role":"sequence","name":"basis_map_A008277 of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","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-646910b0cfed5bbd","kind":"fingerprint_related","node_a_id":"n-07823287c2480f2b","node_b_id":"n-aea9a4160a820c9e","fingerprint":"f6-acf6020f9423e8c7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-acf6020f9423e8c7"},"other_id":"n-07823287c2480f2b","other":{"id":"n-07823287c2480f2b","name":"theta_plus_one_squared of basis_map_A008277 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","first_index":1,"fingerprint":"f6-acf6020f9423e8c7","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","4"]],"card_rows":[["1"],["1"],["2","4"],["3","8"]],"row_sums_preview":["1","1","6","11","38","87","257","647"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8b3f39d03de798f","steps":["basis A008277","(\u03b8+1)\u00b2"],"edges":["e-658de443ccafc3cd","e-79657aa6ed66b72b"],"text":"A052553 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2"},"headline":"A052553 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2","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:  2,    4;\n  3:  3,    8;\n  4:  5,   24,    9;\n  5:  8,   52,   27;\n  6: 13,  120,  108,   16;\n  7: 21,  256,  306,   64;\n  8: 34,  548,  900,  320,  25;\n  9: 55, 1144, 2394, 1120, 125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*x-9*x^2)*P(n-4,x) - (2*x-4*x^2-4*x^3)*d/dx P(n-4,x) - (x^2-4*x^3)*d^2/dx^2 P(n-4,x) - (1+4*x)*P(n-3,x) - (2*x-4*x^2)*d/dx P(n-3,x) + 4*x*P(n-2,x) + (2*x-4*x^2)*d/dx P(n-2,x) + 2*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 2 + 4*x, P(3,x) = 3 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 2 + 4*x;\nP[3, x_] := 3 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-(4*x - 9*x^2)*P[n - 4, x] - (2*x - 4*x^2 - 4*x^3)*D[P[n - 4, x], x] - (x^2 - 4*x^3)*D[P[n - 4, x], {x, 2}] - (1 + 4*x)*P[n - 3, x] - (2*x - 4*x^2)*D[P[n - 3, x], x] + 4*x*P[n - 2, x] + (2*x - 4*x^2)*D[P[n - 2, x], x] + 2*P[n - 1, 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; v[3] = 2 + 4*x; v[4] = 3 + 8*x; for(n = 4, N, my(i = n + 1); v[i] = -(4*x - 9*x^2)*v[i - 4] - (2*x - 4*x^2 - 4*x^3)*deriv(v[i - 4], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 4], x), x) - (1 + 4*x)*v[i - 3] - (2*x - 4*x^2)*deriv(v[i - 3], x) + 4*x*v[i - 2] + (2*x - 4*x^2)*deriv(v[i - 2], x) + 2*v[i - 1]); 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-75ccd0453a9b6cec","steps":["basis A008277","(\u03b8+1)\u00b2"],"edges":["e-67e9a817b66ff47d","e-cbd7f1509507a974"],"text":"A011973 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2"},"headline":"A011973 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2","chain":null}
