{"id":"n-fb16da45c5e69d3d","revision":1,"first_index":0,"fingerprint":"f6-127028c66ab36cef","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","2"],["4","40","4"],["8","472","472","8"],["16","4928","16992","4928","16"],["32","49824","433984","433984","49824","32"],["64","499584","9505728","22567168","9505728","499584","64"],["128","4999040","192085632","909941120","909941120","192085632","4999040","128"],["256","49997824","3707493376","31726065664","61875996160","31726065664","3707493376","49997824","256"],["512","499995136","69634754560","1010252376064","3436278627328","3436278627328","1010252376064","69634754560","499995136","512"],["1024","4999989248","1286425261056","30305377542144","167346092132352","288647404695552","167346092132352","30305377542144","1286425261056","4999989248","1024"],["2048","49999976448","23525653903360","872843883325440","7447479029944320","20490495603830784","20490495603830784","7447479029944320","872843883325440","23525653903360","49999976448","2048"],["4096","499999948800","427561768329216","24434839355310080","310821983317585920","1292554599097663488","2049049560383078400","1292554599097663488","310821983317585920","24434839355310080","427561768329216","499999948800","4096"],["8192","4999999889408","7741111825317888","670365888241057792","12376125545090867200","74801544061062537216","177420644766818402304","177420644766818402304","74801544061062537216","12376125545090867200","670365888241057792","7741111825317888","4999999889408","8192"],["16384","49999999762432","139830012844883968","18126213158546112512","475758271368856223744","4057498140901350735872","13807934146371047571456","20580794792950934667264","13807934146371047571456","4057498140901350735872","475758271368856223744","18126213158546112512","139830012844883968","49999999762432","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","48","960","26880","967680","42577920","2214051840","132843110400","9033331507200","686533194547200","57668788341964800","5305528527460761600","530552852746076160000","57299708096576225280000"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A257618","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","-6"],["0","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159768,"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":false,"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-9fe79b4a02dbc9c6"]},{"id":159769,"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-9fe79b4a02dbc9c6"]},{"id":159770,"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-9fe79b4a02dbc9c6"]},{"id":159771,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-9fe79b4a02dbc9c6"]}],"region_exact":{},"incoming_edges":[{"id":"e-9fe79b4a02dbc9c6","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-fb16da45c5e69d3d","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"0716b628d5949e43aabd48742392de54b5601bcd989f10ae085d90e3247a5c5c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-9580ca662e76a47a","operation":"gamma_polynomial","operation_detail":"scale 1, shift 0","conclusion_id":"n-a71082a9c4388af6","conclusion_name":"gamma_polynomial of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2."},{"id":"e-3a6b6c46f0fb93e5","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-e81ba8f8cf6cbd29","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2."},{"id":"e-3a6b6c46f0fb93e5","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-e81ba8f8cf6cbd29","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2."}],"used_by_total":3,"links":[{"id":"l-7f69ff309d350a23","kind":"fingerprint_related","node_a_id":"n-d023ddeb9da11712","node_b_id":"n-fb16da45c5e69d3d","fingerprint":"f6-127028c66ab36cef","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-127028c66ab36cef"},"other_id":"n-d023ddeb9da11712","other":{"id":"n-d023ddeb9da11712","name":"Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).","first_index":1,"fingerprint":"f6-127028c66ab36cef","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","10","1"]],"card_rows":[["1"],["1","1"],["1","10","1"],["1","59","59","1"]],"row_sums_preview":["1","2","12","120","1680","30240","665280","17297280"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A142459"],"derivation":{"root":"n-d023ddeb9da11712","steps":[],"edges":[],"text":"A142459"},"headline":"A142459: Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).","chain":null}},{"id":"l-1d5b7804c3bd0729","kind":"oeis_match","node_a_id":"n-fb16da45c5e69d3d","node_b_id":null,"fingerprint":null,"oeis_id":"A052714","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":"A052714","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-976ef7dae05e1c30","kind":"oeis_match","node_a_id":"n-fb16da45c5e69d3d","node_b_id":null,"fingerprint":null,"oeis_id":"A144828","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":"A144828","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f469fc4e46a96f10","kind":"oeis_match","node_a_id":"n-fb16da45c5e69d3d","node_b_id":null,"fingerprint":null,"oeis_id":"A257618","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":36},"in_catalog":true,"oeis_id":"A257618","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fc13285f8f2213d5","kind":"oeis_match","node_a_id":"n-fb16da45c5e69d3d","node_b_id":null,"fingerprint":null,"oeis_id":"A257618","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"reversed_triangle","terms_compared":36},"in_catalog":true,"oeis_id":"A257618","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A257618","relation":"identification"},{"oeis_id":"A257618","relation":"match_reversed_triangle"},{"oeis_id":"A052714","relation":"match_row_sums"},{"oeis_id":"A144828","relation":"match_row_sums"},{"oeis_id":"A257618","relation":"match_triangle"}],"bfiles":[{"node_id":"n-fb16da45c5e69d3d","oeis_id":"A144828","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":18,"oeis_terms_available":18,"term_count":200,"raw_size":51792},{"node_id":"n-fb16da45c5e69d3d","oeis_id":"A257618","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":36,"oeis_terms_agreeing":36,"oeis_terms_available":36,"term_count":990,"raw_size":50640},{"node_id":"n-fb16da45c5e69d3d","oeis_id":"A257618","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":36,"oeis_terms_agreeing":36,"oeis_terms_available":36,"term_count":990,"raw_size":50640}],"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,        40,           4;\n  3:   8,       472,         472,             8;\n  4:  16,      4928,       16992,          4928,            16;\n  5:  32,     49824,      433984,        433984,         49824,            32;\n  6:  64,    499584,     9505728,      22567168,       9505728,        499584,            64;\n  7: 128,   4999040,   192085632,     909941120,     909941120,     192085632,       4999040,         128;\n  8: 256,  49997824,  3707493376,   31726065664,   61875996160,   31726065664,    3707493376,    49997824,       256;\n  9: 512, 499995136, 69634754560, 1010252376064, 3436278627328, 3436278627328, 1010252376064, 69634754560, 499995136, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(8*n-6)*x)*P(n-1,x) + (8*x-8*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[(2 + (8*n - 6)*x)*P[n - 1, x] + (8*x - 8*x^2)*D[P[n - 1, x], 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] = (2 + (8*n - 6)*x)*v[i - 1] + (8*x - 8*x^2)*deriv(v[i - 1], 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}},{"oeis_id":"A257618","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   2,         2;\n  2:   4,        40,           4;\n  3:   8,       472,         472,             8;\n  4:  16,      4928,       16992,          4928,            16;\n  5:  32,     49824,      433984,        433984,         49824,            32;\n  6:  64,    499584,     9505728,      22567168,       9505728,        499584,            64;\n  7: 128,   4999040,   192085632,     909941120,     909941120,     192085632,       4999040,         128;\n  8: 256,  49997824,  3707493376,   31726065664,   61875996160,   31726065664,    3707493376,    49997824,       256;\n  9: 512, 499995136, 69634754560, 1010252376064, 3436278627328, 3436278627328, 1010252376064, 69634754560, 499995136, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(8*n-6)*x)*P(n-1,x) + (8*x-8*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[(2 + (8*n - 6)*x)*P[n - 1, x] + (8*x - 8*x^2)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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 + (8*n - 6)*x)*v[i - 1] + (8*x - 8*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}},{"oeis_id":"A257618","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   2,         2;\n  2:   4,        40,           4;\n  3:   8,       472,         472,             8;\n  4:  16,      4928,       16992,          4928,            16;\n  5:  32,     49824,      433984,        433984,         49824,            32;\n  6:  64,    499584,     9505728,      22567168,       9505728,        499584,            64;\n  7: 128,   4999040,   192085632,     909941120,     909941120,     192085632,       4999040,         128;\n  8: 256,  49997824,  3707493376,   31726065664,   61875996160,   31726065664,    3707493376,    49997824,       256;\n  9: 512, 499995136, 69634754560, 1010252376064, 3436278627328, 3436278627328, 1010252376064, 69634754560, 499995136, 512;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+(8*n-6)*x)*P(n-1,x) + (8*x-8*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[(2 + (8*n - 6)*x)*P[n - 1, x] + (8*x - 8*x^2)*D[P[n - 1, x], x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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 + (8*n - 6)*x)*v[i - 1] + (8*x - 8*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-fb16da45c5e69d3d","steps":[],"edges":[],"text":"A257618"},"headline":"A257618: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.","chain":null}