{"id":"n-9f865b2dcffd9592","revision":1,"first_index":0,"fingerprint":"f6-6b183ed9136cbe2a","name":"theta_plus_one of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["1","2"],["0","0","3"],["0","0","3","4"],["0","0","0","0","5"],["0","0","0","0","5","6"],["0","0","0","0","0","0","7"],["0","0","0","0","0","0","7","8"],["0","0","0","0","0","0","0","0","9"],["0","0","0","0","0","0","0","0","9","10"],["0","0","0","0","0","0","0","0","0","0","11"],["0","0","0","0","0","0","0","0","0","0","11","12"],["0","0","0","0","0","0","0","0","0","0","0","0","13"],["0","0","0","0","0","0","0","0","0","0","0","0","13","14"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","3","7","5","11","7","15","9","19","11","23","13","27","15"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-2"],["-2","1"],["3","3"],["-1","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"],["0","-1","0"],["0","1","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160670,"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-246a0576b6d0a23b"]},{"id":160671,"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-246a0576b6d0a23b"]},{"id":160672,"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-246a0576b6d0a23b"]}],"region_exact":{},"incoming_edges":[{"id":"e-246a0576b6d0a23b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-9f865b2dcffd9592","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":"20668a238db4af0f9afe523efa572c44d37c2e5f097f398d1ca5cbeed8b197a5","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-ea2139b8dbf6bf8d","role":"sequence","name":"Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","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-9b8c48ebb8d8d593","kind":"oeis_match","node_a_id":"n-9f865b2dcffd9592","node_b_id":null,"fingerprint":null,"oeis_id":"A114753","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A114753","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A114753","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-9f865b2dcffd9592","oeis_id":"A114753","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":69,"oeis_terms_available":69,"term_count":200,"raw_size":1410}],"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, 2;\n  2: 0, 0, 3;\n  3: 0, 0, 3, 4;\n  4: 0, 0, 0, 0, 5;\n  5: 0, 0, 0, 0, 5, 6;\n  6: 0, 0, 0, 0, 0, 0, 7;\n  7: 0, 0, 0, 0, 0, 0, 7, 8;\n  8: 0, 0, 0, 0, 0, 0, 0, 0, 9;\n  9: 0, 0, 0, 0, 0, 0, 0, 0, 9, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n)*P(n,x) = -(n^3-3*n^2+2*n+(2*n^3-3*n^2-n+2)*x)*P(n-1,x) + ((n^2-n)*x+(2*n^2-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 3*n^2 + 2*n + (2*n^3 - 3*n^2 - n + 2)*x)*P[n - 1, x] + ((n^2 - n)*x + (2*n^2 - 1)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 3*n^2 + 2*n + (2*n^3 - 3*n^2 - n + 2)*x)*v[i - 1] + ((n^2 - n)*x + (2*n^2 - 1)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); 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-ea2139b8dbf6bf8d","steps":["\u03b8+1"],"edges":["e-246a0576b6d0a23b"],"text":"A133080 \u2192 \u03b8+1"},"headline":"A133080 \u2192 \u03b8+1","chain":null}