{"id":"n-f8bb6517bc8332de","revision":1,"first_index":0,"fingerprint":"f6-5410ddd5b5edb7cf","name":"theta_plus_one of A Jacobsthal triangle.","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":false,"rows":[["1"],["0","2"],["0","4","3"],["0","0","12","4"],["0","0","12","24","5"],["0","0","0","48","40","6"],["0","0","0","32","120","60","7"],["0","0","0","0","160","240","84","8"],["0","0","0","0","80","480","420","112","9"],["0","0","0","0","0","480","1120","672","144","10"],["0","0","0","0","0","192","1680","2240","1008","180","11"],["0","0","0","0","0","0","1344","4480","4032","1440","220","12"],["0","0","0","0","0","0","448","5376","10080","6720","1980","264","13"],["0","0","0","0","0","0","0","3584","16128","20160","10560","2640","312","14"],["0","0","0","0","0","0","0","1024","16128","40320","36960","15840","3432","364","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","7","16","41","94","219","492","1101","2426","5311","11528","24881","53398","114083"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","2"],["-4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161117,"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-ecfefb0c627755a3"]},{"id":161118,"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-ecfefb0c627755a3"]},{"id":161119,"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-ecfefb0c627755a3"]}],"region_exact":{},"incoming_edges":[{"id":"e-ecfefb0c627755a3","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-f8bb6517bc8332de","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":"44af763d789661426bfa774a88d29bd2192e2e564cbf07ce083ffcb707c001b5","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-ba196dcd01800741","role":"sequence","name":"A Jacobsthal triangle.","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-4da589cf453d4b8c","kind":"fingerprint_related","node_a_id":"n-f8bb6517bc8332de","node_b_id":"n-f8f37b87137bc296","fingerprint":"f6-5410ddd5b5edb7cf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5410ddd5b5edb7cf"},"other_id":"n-f8f37b87137bc296","other":{"id":"n-f8f37b87137bc296","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)).","first_index":0,"fingerprint":"f6-5410ddd5b5edb7cf","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["4"],["6","8"],["8","24"]],"row_sums_preview":["0","0","2","4","14","32","82","188"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e3d901814302c059","steps":["D"],"edges":["e-781b6acf5129326e"],"text":"A128099 \u2192 D"},"headline":"A128099 \u2192 D","chain":null}},{"id":"l-525b94a819c7167f","kind":"oeis_match","node_a_id":"n-f8bb6517bc8332de","node_b_id":null,"fingerprint":null,"oeis_id":"A073371","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":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A073371","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A073371","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-f8bb6517bc8332de","oeis_id":"A073371","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":30,"oeis_terms_available":30,"term_count":200,"raw_size":7288}],"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: 0, 2;\n  2: 0, 4,  3;\n  3: 0, 0, 12,  4;\n  4: 0, 0, 12, 24,   5;\n  5: 0, 0,  0, 48,  40,   6;\n  6: 0, 0,  0, 32, 120,  60,    7;\n  7: 0, 0,  0,  0, 160, 240,   84,   8;\n  8: 0, 0,  0,  0,  80, 480,  420, 112,   9;\n  9: 0, 0,  0,  0,   0, 480, 1120, 672, 144, 10;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(4*n-4-2*x)*P(n-1,x) + (8*x+x^2)*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[(-(4*n - 4 - 2*x)*P[n - 1, x] + (8*x + x^2)*D[P[n - 1, x], x])/(n)]];\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] = (-(4*n - 4 - 2*x)*v[i - 1] + (8*x + x^2)*deriv(v[i - 1], x))/(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-ba196dcd01800741","steps":["\u03b8+1"],"edges":["e-ecfefb0c627755a3"],"text":"A113953 \u2192 \u03b8+1"},"headline":"A113953 \u2192 \u03b8+1","chain":null}