{"id":"n-8ce0c552efe4fe6e","revision":1,"first_index":0,"fingerprint":"f6-2f0e99ca669c600a","name":"theta_plus_one of Riordan array ((1+x)^3,x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1","6","9","4"],["0","2","9","12","5"],["0","0","3","12","15","6"],["0","0","0","4","15","18","7"],["0","0","0","0","5","18","21","8"],["0","0","0","0","0","6","21","24","9"],["0","0","0","0","0","0","7","24","27","10"],["0","0","0","0","0","0","0","8","27","30","11"],["0","0","0","0","0","0","0","0","9","30","33","12"],["0","0","0","0","0","0","0","0","0","10","33","36","13"],["0","0","0","0","0","0","0","0","0","0","11","36","39","14"],["0","0","0","0","0","0","0","0","0","0","0","12","39","42","15"],["0","0","0","0","0","0","0","0","0","0","0","0","13","42","45","16"],["0","0","0","0","0","0","0","0","0","0","0","0","0","14","45","48","17"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","15","48","51","18"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["20","28","36","44","52","60","68","76","84","92","100","108","116","124","132"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","6","9","4"]],"recurrence":{"denominator":{"coeffs":[["0","3"],["1","1"]]},"terms":[{"coeff":{"coeffs":[["0","1","4"],["0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160688,"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-c8839bd61c0ac6a5"]},{"id":160689,"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-c8839bd61c0ac6a5"]},{"id":160690,"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-c8839bd61c0ac6a5"]}],"region_exact":{},"incoming_edges":[{"id":"e-c8839bd61c0ac6a5","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-8ce0c552efe4fe6e","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":"cf6162056b6a7ec9c6ebbb49ac0fcd1f5dc9075a760971044f152c7cd1bfc1eb","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-e1d160f3b707da31","role":"sequence","name":"Riordan array ((1+x)^3,x).","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-39cfaccd1d77f18b","kind":"oeis_match","node_a_id":"n-8ce0c552efe4fe6e","node_b_id":null,"fingerprint":null,"oeis_id":"A017113","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A017113","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-7e4ea1e39546ba87","kind":"oeis_match","node_a_id":"n-8ce0c552efe4fe6e","node_b_id":null,"fingerprint":null,"oeis_id":"A086570","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A086570","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-7f3ae9b2e309c800","kind":"oeis_match","node_a_id":"n-8ce0c552efe4fe6e","node_b_id":null,"fingerprint":null,"oeis_id":"A136725","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":3,"target":"row_sums","terms_compared":11},"in_catalog":false,"oeis_id":"A136725","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a36555c86952921f","kind":"oeis_match","node_a_id":"n-8ce0c552efe4fe6e","node_b_id":null,"fingerprint":null,"oeis_id":"A281813","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A281813","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A017113","relation":"match_row_sums"},{"oeis_id":"A086570","relation":"match_row_sums"},{"oeis_id":"A136725","relation":"match_row_sums"},{"oeis_id":"A281813","relation":"match_row_sums"}],"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, 6, 9,  4;\n  1: 0, 2, 9, 12,  5;\n  2: 0, 0, 3, 12, 15,  6;\n  3: 0, 0, 0,  4, 15, 18,  7;\n  4: 0, 0, 0,  0,  5, 18, 21,  8;\n  5: 0, 0, 0,  0,  0,  6, 21, 24,  9;\n  6: 0, 0, 0,  0,  0,  0,  7, 24, 27, 10;\n  7: 0, 0, 0,  0,  0,  0,  0,  8, 27, 30, 11;\n  8: 0, 0, 0,  0,  0,  0,  0,  0,  9, 30, 33, 12;\n  9: 0, 0, 0,  0,  0,  0,  0,  0,  0, 10, 33, 36, 13;\n","recurrence_sentence":"The row polynomials satisfy (n+(n+3)*x)*P(n,x) = ((n+1)*x+(n+4)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 6*x + 9*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1 + 6*x + 9*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(((n + 1)*x + (n + 4)*x^2)*P[n - 1, x])/(n + (n + 3)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 6*x + 9*x^2 + 4*x^3; for(n = 1, N, my(i = n + 1); v[i] = (((n + 1)*x + (n + 4)*x^2)*v[i - 1])/(n + (n + 3)*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-e1d160f3b707da31","steps":["\u03b8+1"],"edges":["e-c8839bd61c0ac6a5"],"text":"A122431 \u2192 \u03b8+1"},"headline":"A122431 \u2192 \u03b8+1","chain":null}