{"id":"n-f3a4ae052af3da90","revision":1,"first_index":0,"fingerprint":"f6-560c70a8d992c842","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) = 3*x + 5.","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"],["5","5"],["25","80","25"],["125","915","915","125"],["625","9070","20130","9070","625"],["3125","83185","348410","348410","83185","3125"],["15625","727980","5246655","9755480","5246655","727980","15625"],["78125","6183215","72272805","225769855","225769855","72272805","6183215","78125"],["390625","51496970","937214800","4606234070","7676175070","4606234070","937214800","51496970","390625"],["1953125","423303885","11648284020","86043217380","222619657590","222619657590","86043217380","11648284020","423303885","1953125"],["9765625","3448931080","140406936885","1507460427840","5763528178770","8904786303600","5763528178770","1507460427840","140406936885","3448931080","9765625"],["48828125","27933245515","1654842100295","25176247159425","137173950162930","310656874183710","310656874183710","137173950162930","25176247159425","1654842100295","27933245515","48828125"],["244140625","225321432870","19180926696270","405422407441390","3062068320393135","9779660187910380","14290216212450660","9779660187910380","3062068320393135","405422407441390","19180926696270","225321432870","244140625"],["1220703125","1812581228585","219552408108030","6347246138548910","65028678484807775","284393185049608515","582946137772035060","582946137772035060","284393185049608515","65028678484807775","6347246138548910","219552408108030","1812581228585","1220703125"],["6103515625","14554360766180","2489392319560315","97204437447789880","1327641149090944025","7768781412506019100","21655163535195453315","30313199164145823120","21655163535195453315","7768781412506019100","1327641149090944025","97204437447789880","2489392319560315","14554360766180","6103515625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","130","2080","39520","869440","21736000","608608000","18866848000","641472832000","23734494784000","949379791360000","40823331028480000","1877873227310080000","92015788138193920000"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A257624","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","2"],["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159793,"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-013c0ad711a68b34"]},{"id":159794,"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-013c0ad711a68b34"]},{"id":159795,"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-013c0ad711a68b34"]},{"id":159796,"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-013c0ad711a68b34"]}],"region_exact":{},"incoming_edges":[{"id":"e-013c0ad711a68b34","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-f3a4ae052af3da90","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":"d3a8d69c84f8d518e5b67a1afaafe93d33d50d5c6ea11c2d1a18c196821b446e","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-7fb7a0f59a4bd766","operation":"gamma_polynomial","operation_detail":"scale 1, shift 0","conclusion_id":"n-ca46250f05be6e7c","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) = 3*x + 5."},{"id":"e-148e9e46774006f0","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-a375f9a732b31b5d","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) = 3*x + 5."},{"id":"e-148e9e46774006f0","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-a375f9a732b31b5d","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) = 3*x + 5."}],"used_by_total":3,"links":[{"id":"l-2030ae2b803c59fa","kind":"oeis_match","node_a_id":"n-f3a4ae052af3da90","node_b_id":null,"fingerprint":null,"oeis_id":"A257624","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":"A257624","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-4b4d0ceaa97f3657","kind":"oeis_match","node_a_id":"n-f3a4ae052af3da90","node_b_id":null,"fingerprint":null,"oeis_id":"A257624","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":"A257624","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-bbd5afa659726429","kind":"oeis_match","node_a_id":"n-f3a4ae052af3da90","node_b_id":null,"fingerprint":null,"oeis_id":"A051607","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":"A051607","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A257624","relation":"identification"},{"oeis_id":"A257624","relation":"match_reversed_triangle"},{"oeis_id":"A051607","relation":"match_row_sums"},{"oeis_id":"A257624","relation":"match_triangle"}],"bfiles":[{"node_id":"n-f3a4ae052af3da90","oeis_id":"A051607","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":44342},{"node_id":"n-f3a4ae052af3da90","oeis_id":"A257624","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":44706},{"node_id":"n-f3a4ae052af3da90","oeis_id":"A257624","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":44706}],"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:       5,         5;\n  2:      25,        80,          25;\n  3:     125,       915,         915,         125;\n  4:     625,      9070,       20130,        9070,          625;\n  5:    3125,     83185,      348410,      348410,        83185,         3125;\n  6:   15625,    727980,     5246655,     9755480,      5246655,       727980,       15625;\n  7:   78125,   6183215,    72272805,   225769855,    225769855,     72272805,     6183215,       78125;\n  8:  390625,  51496970,   937214800,  4606234070,   7676175070,   4606234070,   937214800,    51496970,    390625;\n  9: 1953125, 423303885, 11648284020, 86043217380, 222619657590, 222619657590, 86043217380, 11648284020, 423303885, 1953125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+(3*n+2)*x)*P(n-1,x) + (3*x-3*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[(5 + (3*n + 2)*x)*P[n - 1, x] + (3*x - 3*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] = (5 + (3*n + 2)*x)*v[i - 1] + (3*x - 3*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":"A257624","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:       5,         5;\n  2:      25,        80,          25;\n  3:     125,       915,         915,         125;\n  4:     625,      9070,       20130,        9070,          625;\n  5:    3125,     83185,      348410,      348410,        83185,         3125;\n  6:   15625,    727980,     5246655,     9755480,      5246655,       727980,       15625;\n  7:   78125,   6183215,    72272805,   225769855,    225769855,     72272805,     6183215,       78125;\n  8:  390625,  51496970,   937214800,  4606234070,   7676175070,   4606234070,   937214800,    51496970,    390625;\n  9: 1953125, 423303885, 11648284020, 86043217380, 222619657590, 222619657590, 86043217380, 11648284020, 423303885, 1953125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+(3*n+2)*x)*P(n-1,x) + (3*x-3*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[(5 + (3*n + 2)*x)*P[n - 1, x] + (3*x - 3*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] = (5 + (3*n + 2)*x)*v[i - 1] + (3*x - 3*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":"A257624","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:       5,         5;\n  2:      25,        80,          25;\n  3:     125,       915,         915,         125;\n  4:     625,      9070,       20130,        9070,          625;\n  5:    3125,     83185,      348410,      348410,        83185,         3125;\n  6:   15625,    727980,     5246655,     9755480,      5246655,       727980,       15625;\n  7:   78125,   6183215,    72272805,   225769855,    225769855,     72272805,     6183215,       78125;\n  8:  390625,  51496970,   937214800,  4606234070,   7676175070,   4606234070,   937214800,    51496970,    390625;\n  9: 1953125, 423303885, 11648284020, 86043217380, 222619657590, 222619657590, 86043217380, 11648284020, 423303885, 1953125;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (5+(3*n+2)*x)*P(n-1,x) + (3*x-3*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[(5 + (3*n + 2)*x)*P[n - 1, x] + (3*x - 3*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] = (5 + (3*n + 2)*x)*v[i - 1] + (3*x - 3*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-f3a4ae052af3da90","steps":[],"edges":[],"text":"A257624"},"headline":"A257624: 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) = 3*x + 5.","chain":null}