{"id":"n-09791955a4a02550","revision":1,"first_index":0,"fingerprint":"f6-e620d4a356abb37c","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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"],["2","2"],["4","8","3"],["8","24","18","4"],["16","64","72","32","5"],["32","160","240","160","50","6"],["64","384","720","640","300","72","7"],["128","896","2016","2240","1400","504","98","8"],["256","2048","5376","7168","5600","2688","784","128","9"],["512","4608","13824","21504","20160","12096","4704","1152","162","10"],["1024","10240","34560","61440","67200","48384","23520","7680","1620","200","11"],["2048","22528","84480","168960","211200","177408","103488","42240","11880","2200","242","12"],["4096","49152","202752","450560","633600","608256","413952","202752","71280","17600","2904","288","13"],["8192","106496","479232","1171456","1830400","1976832","1537536","878592","370656","114400","25168","3744","338","14"],["16384","229376","1118208","2981888","5125120","6150144","5381376","3514368","1729728","640640","176176","34944","4732","392","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","15","54","189","648","2187","7290","24057","78732","255879","826686","2657205","8503056","27103491"],"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":[["2","0"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["4","4","1"],["0","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163701,"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-78274d1fdc91ece9"]},{"id":163702,"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-78274d1fdc91ece9"]},{"id":163703,"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-78274d1fdc91ece9"]},{"id":163704,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-78274d1fdc91ece9"]}],"region_exact":{},"incoming_edges":[{"id":"e-78274d1fdc91ece9","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-09791955a4a02550","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":"404792e0bb514ba9dd3783fccf4c744d3b5073fb67e6ccb324c6765827e73ec2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-af754f07daf65b5e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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-51500dfbc7a95314","kind":"fingerprint_related","node_a_id":"n-09791955a4a02550","node_b_id":"n-4ec97bc875f9c45e","fingerprint":"f6-e620d4a356abb37c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e620d4a356abb37c"},"other_id":"n-4ec97bc875f9c45e","other":{"id":"n-4ec97bc875f9c45e","name":"theta_plus_one of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-e620d4a356abb37c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["6","6"],["12","24","9"]],"card_rows":[["3"],["6","6"],["12","24","9"],["24","72","54","12"]],"row_sums_preview":["3","12","45","162","567","1944","6561","21870"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["\u03b8+1"],"edges":["e-217a3f649e225959"],"text":"A120910 \u2192 \u03b8+1"},"headline":"A120910 \u2192 \u03b8+1","chain":null}},{"id":"l-6384871c7516d346","kind":"oeis_match","node_a_id":"n-09791955a4a02550","node_b_id":null,"fingerprint":null,"oeis_id":"A006234","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":3,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A006234","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A006234","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-09791955a4a02550","oeis_id":"A006234","target":"row_sums","oeis_offset":3,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":10770}],"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:   2,    2;\n  2:   4,    8,     3;\n  3:   8,   24,    18,     4;\n  4:  16,   64,    72,    32,     5;\n  5:  32,  160,   240,   160,    50,     6;\n  6:  64,  384,   720,   640,   300,    72,    7;\n  7: 128,  896,  2016,  2240,  1400,   504,   98,    8;\n  8: 256, 2048,  5376,  7168,  5600,  2688,  784,  128,   9;\n  9: 512, 4608, 13824, 21504, 20160, 12096, 4704, 1152, 162, 10;\n","recurrence_sentence":"The row polynomials satisfy (2+n*x)*P(n,x) = (4+(2*n+4)*x+(n+1)*x^2)*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 + (2*n + 4)*x + (n + 1)*x^2)*P[n - 1, x])/(2 + n*x)]];\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 + (2*n + 4)*x + (n + 1)*x^2)*v[i - 1])/(2 + n*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-af754f07daf65b5e","steps":["\u03b8+1"],"edges":["e-78274d1fdc91ece9"],"text":"A038207 \u2192 \u03b8+1"},"headline":"A038207 \u2192 \u03b8+1","chain":null}
