{"id":"n-d023ddeb9da11712","revision":1,"first_index":1,"fingerprint":"f6-127028c66ab36cef","name":"Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","1"],["1","10","1"],["1","59","59","1"],["1","308","1062","308","1"],["1","1557","13562","13562","1557","1"],["1","7806","148527","352612","148527","7806","1"],["1","39055","1500669","7108915","7108915","1500669","39055","1"],["1","195304","14482396","123929944","241703110","123929944","14482396","195304","1"],["1","976553","136005380","1973149172","6711481694","6711481694","1973149172","136005380","976553","1"],["1","4882802","1256274669","29595095256","163423918098","281882231148","163423918098","29595095256","1256274669","4882802","1"],["1","24414051","11487135695","426193302405","3636464370090","10005124806558","10005124806558","3636464370090","426193302405","11487135695","24414051","1"],["1","122070300","104385197346","5965536951980","75884273270895","315565087670328","500256240327900","315565087670328","75884273270895","5965536951980","104385197346","122070300","1"],["1","610351549","944959939614","81831773466926","1510757512828475","9131047859016423","21657793550637012","21657793550637012","9131047859016423","1510757512828475","81831773466926","944959939614","610351549","1"],["1","3051757798","8534546682427","1106336252352668","29037980430228041","247650033013998458","842769418113467259","1256152025936946696","842769418113467259","247650033013998458","29037980430228041","1106336252352668","8534546682427","3051757798","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","12","120","1680","30240","665280","17297280","518918400","17643225600","670442572800","28158588057600","1295295050649600","64764752532480000","3497296636753920000"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A142459","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-7"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159288,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0)","region_exact":null,"edges":["e-ce13bfb75f31b51b"]},{"id":159289,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-ce13bfb75f31b51b"]},{"id":159290,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-ce13bfb75f31b51b"]},{"id":159291,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"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 >= 2","region_exact":null,"edges":["e-ce13bfb75f31b51b"]}],"region_exact":{},"incoming_edges":[{"id":"e-ce13bfb75f31b51b","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-d023ddeb9da11712","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":"63b1c07ae2beeacd9fde12c68222b69cd7bb0b4230a3f4a1927196abf87a60d6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-e113043a8d357c9d","operation":"gamma_polynomial","operation_detail":"scale 1, shift 0","conclusion_id":"n-ada634934a2d63b9","conclusion_name":"gamma_polynomial of Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k)."},{"id":"e-2762452c13f28e99","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-f77168e603d1499e","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k)."},{"id":"e-2762452c13f28e99","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-f77168e603d1499e","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k)."}],"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-fb16da45c5e69d3d","other":{"id":"n-fb16da45c5e69d3d","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.","first_index":0,"fingerprint":"f6-127028c66ab36cef","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["4","40","4"]],"card_rows":[["1"],["2","2"],["4","40","4"],["8","472","472","8"]],"row_sums_preview":["1","4","48","960","26880","967680","42577920","2214051840"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A257618"],"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}},{"id":"l-20dbea11228f89ec","kind":"oeis_match","node_a_id":"n-d023ddeb9da11712","node_b_id":null,"fingerprint":null,"oeis_id":"A142459","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"reversed_triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A142459","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-48edf1261de9c22d","kind":"oeis_match","node_a_id":"n-d023ddeb9da11712","node_b_id":null,"fingerprint":null,"oeis_id":"A001813","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001813","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-671bb2790ca1c431","kind":"oeis_match","node_a_id":"n-d023ddeb9da11712","node_b_id":null,"fingerprint":null,"oeis_id":"A142459","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A142459","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A142459","relation":"identification"},{"oeis_id":"A142459","relation":"match_reversed_triangle"},{"oeis_id":"A001813","relation":"match_row_sums"},{"oeis_id":"A142459","relation":"match_triangle"}],"bfiles":[{"node_id":"n-d023ddeb9da11712","oeis_id":"A001813","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":19,"oeis_terms_available":19,"term_count":200,"raw_size":45809},{"node_id":"n-d023ddeb9da11712","oeis_id":"A142459","target":"triangle","oeis_offset":1,"first_row":1,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":42161},{"node_id":"n-d023ddeb9da11712","oeis_id":"A142459","target":"reversed_triangle","oeis_offset":1,"first_row":1,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":42161}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,      1;\n   3: 1,     10,         1;\n   4: 1,     59,        59,          1;\n   5: 1,    308,      1062,        308,          1;\n   6: 1,   1557,     13562,      13562,       1557,          1;\n   7: 1,   7806,    148527,     352612,     148527,       7806,          1;\n   8: 1,  39055,   1500669,    7108915,    7108915,    1500669,      39055,         1;\n   9: 1, 195304,  14482396,  123929944,  241703110,  123929944,   14482396,    195304,      1;\n  10: 1, 976553, 136005380, 1973149172, 6711481694, 6711481694, 1973149172, 136005380, 976553, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(4*n-7)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (4*n - 7)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (4*n - 7)*x)*v[i - 1] + (4*x - 4*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A142459","target":"triangle","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,      1;\n   3: 1,     10,         1;\n   4: 1,     59,        59,          1;\n   5: 1,    308,      1062,        308,          1;\n   6: 1,   1557,     13562,      13562,       1557,          1;\n   7: 1,   7806,    148527,     352612,     148527,       7806,          1;\n   8: 1,  39055,   1500669,    7108915,    7108915,    1500669,      39055,         1;\n   9: 1, 195304,  14482396,  123929944,  241703110,  123929944,   14482396,    195304,      1;\n  10: 1, 976553, 136005380, 1973149172, 6711481694, 6711481694, 1973149172, 136005380, 976553, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(4*n-7)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (4*n - 7)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (4*n - 7)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A142459","target":"reversed_triangle","first_row":1,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,      1;\n   3: 1,     10,         1;\n   4: 1,     59,        59,          1;\n   5: 1,    308,      1062,        308,          1;\n   6: 1,   1557,     13562,      13562,       1557,          1;\n   7: 1,   7806,    148527,     352612,     148527,       7806,          1;\n   8: 1,  39055,   1500669,    7108915,    7108915,    1500669,      39055,         1;\n   9: 1, 195304,  14482396,  123929944,  241703110,  123929944,   14482396,    195304,      1;\n  10: 1, 976553, 136005380, 1973149172, 6711481694, 6711481694, 1973149172, 136005380, 976553, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(4*n-7)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (4*n - 7)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (4*n - 7)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"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}