{"id":"n-723ec2078be12613","revision":1,"first_index":0,"fingerprint":"f6-7729bd9220ade5d3","name":"Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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"],["1","1"],["1","14","1"],["1","83","83","1"],["1","432","1550","432","1"],["1","2181","19898","19898","2181","1"],["1","10930","217887","523548","217887","10930","1"],["1","54679","2199237","10589795","10589795","2199237","54679","1"],["1","273428","21203828","184722860","362147222","184722860","21203828","273428","1"],["1","1367177","198982580","2940289828","10078042014","10078042014","2940289828","198982580","1367177","1"],["1","6835926","1837053773","44079077896","245572851378","424726353476","245572851378","44079077896","1837053773","6835926","1"],["1","34179675","16791881927","634446717477","5464792891722","15098886875502","15098886875502","5464792891722","634446717477","16791881927","34179675","1"],["1","170898424","152555647722","8876455173592","114014537147599","476537909650992","756643249189004","476537909650992","114014537147599","8876455173592","152555647722","170898424","1"],["1","854492173","1380827977278","121715866341006","2269213759801995","13791635000108487","32796076157105876","32796076157105876","13791635000108487","2269213759801995","121715866341006","1380827977278","854492173","1"],["1","4272460922","12470005505675","1645053744001476","43602490257309529","374041302263542438","1276932010569830939","1905198990108896824","1276932010569830939","374041302263542438","43602490257309529","1645053744001476","12470005505675","4272460922","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","16","168","2416","44160","981184","25687424","774547456","26437363200","1007717991424","42429905092608","1955806506225664","97960044931653632","5297665652334198784"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A144441","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-3"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159364,"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-494af060e65fc342"]},{"id":159365,"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-494af060e65fc342"]},{"id":159366,"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-494af060e65fc342"]},{"id":159367,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-494af060e65fc342"]}],"region_exact":{},"incoming_edges":[{"id":"e-494af060e65fc342","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-723ec2078be12613","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 the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":"halfGrowth","lean_declarations":["RealRooted.derivLag_strictInterl_of_nonnegCoeffs","RealRooted.derivLag_strictInterl_and_noCommonRoot","RealRooted.derivLag_eval_zero_pos"],"certificate_hash":"f68b329992fbe4759b11f3375f4d91c53488862995941ac66f01e59353f80f9d","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":"strict","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-24cd925870a5c300","operation":"gamma_polynomial","operation_detail":"scale 1, shift 0","conclusion_id":"n-273eab41f1e8cbee","conclusion_name":"gamma_polynomial of Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1."}],"used_by_total":1,"links":[{"id":"l-3d5bbeff91c8ad63","kind":"oeis_match","node_a_id":"n-723ec2078be12613","node_b_id":null,"fingerprint":null,"oeis_id":"A144441","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"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":"A144441","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-8e873aa997d16c96","kind":"oeis_match","node_a_id":"n-723ec2078be12613","node_b_id":null,"fingerprint":null,"oeis_id":"A144441","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"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":"A144441","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A144441","relation":"identification"},{"oeis_id":"A144441","relation":"match_reversed_triangle"},{"oeis_id":"A144441","relation":"match_triangle"}],"bfiles":[{"node_id":"n-723ec2078be12613","oeis_id":"A144441","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":42312},{"node_id":"n-723ec2078be12613","oeis_id":"A144441","target":"reversed_triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":42312}],"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: 1,       1;\n  2: 1,      14,         1;\n  3: 1,      83,        83,          1;\n  4: 1,     432,      1550,        432,           1;\n  5: 1,    2181,     19898,      19898,        2181,           1;\n  6: 1,   10930,    217887,     523548,      217887,       10930,          1;\n  7: 1,   54679,   2199237,   10589795,    10589795,     2199237,      54679,         1;\n  8: 1,  273428,  21203828,  184722860,   362147222,   184722860,   21203828,    273428,       1;\n  9: 1, 1367177, 198982580, 2940289828, 10078042014, 10078042014, 2940289828, 198982580, 1367177, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(4*n-3)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) + 4*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + (4*n - 3)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x] + 4*x*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + (4*n - 3)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x) + 4*x*v[i - 2]); 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":"A144441","target":"triangle","first_row":0,"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,      14,         1;\n   4: 1,      83,        83,          1;\n   5: 1,     432,      1550,        432,           1;\n   6: 1,    2181,     19898,      19898,        2181,           1;\n   7: 1,   10930,    217887,     523548,      217887,       10930,          1;\n   8: 1,   54679,   2199237,   10589795,    10589795,     2199237,      54679,         1;\n   9: 1,  273428,  21203828,  184722860,   362147222,   184722860,   21203828,    273428,       1;\n  10: 1, 1367177, 198982580, 2940289828, 10078042014, 10078042014, 2940289828, 198982580, 1367177, 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) + 4*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\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] + 4*x*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (4*n - 7)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x) + 4*x*v[i - 2]); 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":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A144441","target":"reversed_triangle","first_row":0,"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,      14,         1;\n   4: 1,      83,        83,          1;\n   5: 1,     432,      1550,        432,           1;\n   6: 1,    2181,     19898,      19898,        2181,           1;\n   7: 1,   10930,    217887,     523548,      217887,       10930,          1;\n   8: 1,   54679,   2199237,   10589795,    10589795,     2199237,      54679,         1;\n   9: 1,  273428,  21203828,  184722860,   362147222,   184722860,   21203828,    273428,       1;\n  10: 1, 1367177, 198982580, 2940289828, 10078042014, 10078042014, 2940289828, 198982580, 1367177, 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) + 4*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\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] + 4*x*P[n - 2, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (4*n - 7)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x) + 4*x*v[i - 2]); 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":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-723ec2078be12613","steps":[],"edges":[],"text":"A144441"},"headline":"A144441: Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","chain":null}