{"id":"n-44248dd6011d9846","revision":0,"first_index":0,"fingerprint":"f6-39dc65612e9c69b1","name":"Riordan array ((1-x)/(1-2*x), x(1-x)/(1-2*x)^2).","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":false,"rows":[["1"],["1","1"],["2","4","1"],["4","13","7","1"],["8","38","33","10","1"],["16","104","129","62","13","1"],["32","272","450","304","100","16","1"],["64","688","1452","1289","590","147","19","1"],["128","1696","4424","4942","2945","1014","203","22","1"],["256","4096","12896","17584","13073","5823","1603","268","25","1"],["512","9728","36288","59040","53154","29364","10409","2384","342","28","1"],["1024","22784","99200","189216","201780","134245","58765","17270","3384","425","31","1"],["2048","52736","264704","583616","724680","568150","298305","107900","27054","4630","517","34","1"],["4096","120832","691968","1743104","2486000","2258520","1392065","602060","185310","40490","6149","618","37","1"],["8192","274432","1777152","5065216","8204768","8522800","6065410","3070400","1127184","301696","58388","7968","728","40","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","7","25","90","325","1175","4250","15375","55625","201250","728125","2634375","9531250","34484375"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A236471","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"],["2","4","1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-8","-2"],["4","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","2"],["-4","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169747,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-4, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-4"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-4, +inf)","region_exact":null,"edges":["e-b3ebc341e0279939"]},{"id":169748,"position":1,"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-b3ebc341e0279939"]}],"region_exact":{},"incoming_edges":[{"id":"e-b3ebc341e0279939","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-44248dd6011d9846","premise_count":0,"theorem":"L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction","lean_route":null,"lean_declarations":["RealRooted.threeTerm_interlaces_of_window"],"certificate_hash":"13e45c161c28a702bbccca9c86ec7b4b9d063e733ef493ab3bb3e471aefe6f2d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-4"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-855f03a28fbf761d","kind":"oeis_match","node_a_id":"n-44248dd6011d9846","node_b_id":null,"fingerprint":null,"oeis_id":"A052936","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":"A052936","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d8b1bdfc63b40c5b","kind":"oeis_match","node_a_id":"n-44248dd6011d9846","node_b_id":null,"fingerprint":null,"oeis_id":"A236471","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":57},"in_catalog":true,"oeis_id":"A236471","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A236471","relation":"identification"},{"oeis_id":"A052936","relation":"match_row_sums"},{"oeis_id":"A236471","relation":"match_triangle"}],"bfiles":[{"node_id":"n-44248dd6011d9846","oeis_id":"A052936","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":12048},{"node_id":"n-44248dd6011d9846","oeis_id":"A236471","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":57,"oeis_terms_agreeing":57,"oeis_terms_available":57,"term_count":990,"raw_size":16241}],"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:   2,    4,     1;\n  3:   4,   13,     7,     1;\n  4:   8,   38,    33,    10,     1;\n  5:  16,  104,   129,    62,    13,    1;\n  6:  32,  272,   450,   304,   100,   16,    1;\n  7:  64,  688,  1452,  1289,   590,  147,   19,   1;\n  8: 128, 1696,  4424,  4942,  2945, 1014,  203,  22,  1;\n  9: 256, 4096, 12896, 17584, 13073, 5823, 1603, 268, 25, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (4*n-8+(n-2)*x)*P(n-1,x) - (4*n-8+(n-2)*x)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 2 + 4*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[2, x_] := 2 + 4*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (n - 2)*x)*P[n - 1, x] - (4*n - 8 + (n - 2)*x)*P[n - 2, x])/(n - 2)]];\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; v[3] = 2 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8 + (n - 2)*x)*v[i - 1] - (4*n - 8 + (n - 2)*x)*v[i - 2])/(n - 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":"A236471","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:   1,    1;\n  2:   2,    4,     1;\n  3:   4,   13,     7,     1;\n  4:   8,   38,    33,    10,     1;\n  5:  16,  104,   129,    62,    13,    1;\n  6:  32,  272,   450,   304,   100,   16,    1;\n  7:  64,  688,  1452,  1289,   590,  147,   19,   1;\n  8: 128, 1696,  4424,  4942,  2945, 1014,  203,  22,  1;\n  9: 256, 4096, 12896, 17584, 13073, 5823, 1603, 268, 25, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (4*n-8+(n-2)*x)*P(n-1,x) - (4*n-8+(n-2)*x)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 2 + 4*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[2, x_] := 2 + 4*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (n - 2)*x)*P[n - 1, x] - (4*n - 8 + (n - 2)*x)*P[n - 2, x])/(n - 2)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; v[3] = 2 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8 + (n - 2)*x)*v[i - 1] - (4*n - 8 + (n - 2)*x)*v[i - 2])/(n - 2)); 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}}],"derivation":{"root":"n-44248dd6011d9846","steps":[],"edges":[],"text":"A236471"},"headline":"A236471: Riordan array ((1-x)/(1-2*x), x(1-x)/(1-2*x)^2).","chain":null}
