{"id":"n-b3afa198ee370ac3","revision":1,"first_index":0,"fingerprint":"f6-75dc0d5334bf7459","name":"theta_plus_one of Riordan array (1, 2+2x).","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","4"],["0","4","12"],["0","0","24","32"],["0","0","12","96","80"],["0","0","0","96","320","192"],["0","0","0","32","480","960","448"],["0","0","0","0","320","1920","2688","1024"],["0","0","0","0","80","1920","6720","7168","2304"],["0","0","0","0","0","960","8960","21504","18432","5120"],["0","0","0","0","0","192","6720","35840","64512","46080","11264"],["0","0","0","0","0","0","2688","35840","129024","184320","112640","24576"],["0","0","0","0","0","0","448","21504","161280","430080","506880","270336","53248"],["0","0","0","0","0","0","0","7168","129024","645120","1351680","1351680","638976","114688"],["0","0","0","0","0","0","0","1024","64512","645120","2365440","4055040","3514368","1490944","245760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","16","56","188","608","1920","5952","18192","54976","164608","489088","1443776","4238336","12382208"],"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":[["2","4"],["-2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163646,"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-d427c42eda759629"]},{"id":163647,"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-d427c42eda759629"]},{"id":163648,"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-d427c42eda759629"]}],"region_exact":{},"incoming_edges":[{"id":"e-d427c42eda759629","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-b3afa198ee370ac3","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":"ecedc6ae9c12484dfb1a8677a6a7554ef9eb11e4e181cf0e8ca30deb0c275650","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-f5a001953ee28e8f","role":"sequence","name":"Riordan array (1, 2+2x).","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-9b6f52c7f03b8fc0","kind":"oeis_match","node_a_id":"n-b3afa198ee370ac3","node_b_id":null,"fingerprint":null,"oeis_id":"A073388","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":"A073388","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ada81fa036eaf7a1","kind":"oeis_match","node_a_id":"n-b3afa198ee370ac3","node_b_id":null,"fingerprint":null,"oeis_id":"A109634","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A109634","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A073388","relation":"match_row_sums"},{"oeis_id":"A109634","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-b3afa198ee370ac3","oeis_id":"A073388","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":10011}],"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, 4;\n  2: 0, 4, 12;\n  3: 0, 0, 24, 32;\n  4: 0, 0, 12, 96,  80;\n  5: 0, 0,  0, 96, 320,  192;\n  6: 0, 0,  0, 32, 480,  960,  448;\n  7: 0, 0,  0,  0, 320, 1920, 2688,  1024;\n  8: 0, 0,  0,  0,  80, 1920, 6720,  7168,  2304;\n  9: 0, 0,  0,  0,   0,  960, 8960, 21504, 18432, 5120;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(2*n-2-4*x)*P(n-1,x) + (4*x+2*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[(-(2*n - 2 - 4*x)*P[n - 1, x] + (4*x + 2*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] = (-(2*n - 2 - 4*x)*v[i - 1] + (4*x + 2*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-f5a001953ee28e8f","steps":["\u03b8+1"],"edges":["e-d427c42eda759629"],"text":"A099040 \u2192 \u03b8+1"},"headline":"A099040 \u2192 \u03b8+1","chain":null}