{"id":"n-eacb751f39dbd056","revision":0,"first_index":0,"fingerprint":"f6-a866a9bf376c404a","name":"Expansion of 1/sqrt(1-4*x*y-4*x^2*y).","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"],["0","2"],["0","2","6"],["0","0","12","20"],["0","0","6","60","70"],["0","0","0","60","280","252"],["0","0","0","20","420","1260","924"],["0","0","0","0","280","2520","5544","3432"],["0","0","0","0","70","2520","13860","24024","12870"],["0","0","0","0","0","1260","18480","72072","102960","48620"],["0","0","0","0","0","252","13860","120120","360360","437580","184756"],["0","0","0","0","0","0","5544","120120","720720","1750320","1847560","705432"],["0","0","0","0","0","0","924","72072","900900","4084080","8314020","7759752","2704156"],["0","0","0","0","0","0","0","24024","720720","6126120","22170720","38798760","32449872","10400600"],["0","0","0","0","0","0","0","3432","360360","6126120","38798760","116396280","178474296","135207800","40116600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","8","32","136","592","2624","11776","53344","243392","1116928","5149696","23835904","110690816","515483648"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A115951","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-2"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4"],["0","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169344,"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":true,"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-7fa6011f01ff7cdd"]},{"id":169345,"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-7fa6011f01ff7cdd"]},{"id":169346,"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-7fa6011f01ff7cdd"]},{"id":169347,"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-7fa6011f01ff7cdd"]}],"region_exact":{},"incoming_edges":[{"id":"e-7fa6011f01ff7cdd","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-eacb751f39dbd056","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":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.threeTerm_interlaces","RealRooted.threeTerm_hasNonnegCoeffs"],"certificate_hash":"ba475ae6345fa172e3dae5effc0b5cbe5c4e671949213d21fe04c84e43df30bb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-1645e47027924141","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-2a67a442e762ff53","conclusion_name":"basis_map_A013610 of Expansion of 1/sqrt(1-4*x*y-4*x^2*y)."},{"id":"e-5ebb49cd3f3bda68","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-68250fd6ae5aea2b","conclusion_name":"basis_map_A013611 of Expansion of 1/sqrt(1-4*x*y-4*x^2*y)."},{"id":"e-7adabb58a59dfd52","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-406d5bfd4a642364","conclusion_name":"basis_map_A013609 of Expansion of 1/sqrt(1-4*x*y-4*x^2*y)."},{"id":"e-7fec2eeeaaa65e45","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-445f9a43ace6155b","conclusion_name":"basis_map_A008277 of Expansion of 1/sqrt(1-4*x*y-4*x^2*y)."},{"id":"e-93ddd9dff3a50248","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-234411cdfee41c66","conclusion_name":"shift_x_plus_one of Expansion of 1/sqrt(1-4*x*y-4*x^2*y)."},{"id":"e-28a30dd28d3eabe7","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-27decca1d071fa2c","conclusion_name":"one_plus_derivative of Expansion of 1/sqrt(1-4*x*y-4*x^2*y)."},{"id":"e-2ba3f457cee339c3","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-8ba32770adecde3b","conclusion_name":"mobius_x_over_x_plus_one of Expansion of 1/sqrt(1-4*x*y-4*x^2*y)."},{"id":"e-2c33fb3ae133abb8","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-8628db06a7bc9bde","conclusion_name":"theta_plus_one_squared of Expansion of 1/sqrt(1-4*x*y-4*x^2*y)."},{"id":"e-43f4e7bf47d18868","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-b2f14d9f44a39df2","conclusion_name":"Triangle of coefficients of Legendre polynomials 2^n P_n (x)."},{"id":"e-95b5d2a6695ffaa1","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-c58260973df006a6","conclusion_name":"theta_plus_one of Expansion of 1/sqrt(1-4*x*y-4*x^2*y)."},{"id":"e-a06a989e5deb8ec0","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-fbfc644b975c7953","conclusion_name":"derivative of Expansion of 1/sqrt(1-4*x*y-4*x^2*y)."},{"id":"e-dc7854e01cf28fd9","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-234411cdfee41c66","conclusion_name":"shift_x_plus_one of Expansion of 1/sqrt(1-4*x*y-4*x^2*y)."}],"used_by_total":12,"links":[{"id":"l-467c3ca43ce19b91","kind":"fingerprint_related","node_a_id":"n-b2f14d9f44a39df2","node_b_id":"n-eacb751f39dbd056","fingerprint":"f6-a866a9bf376c404a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a866a9bf376c404a"},"other_id":"n-b2f14d9f44a39df2","other":{"id":"n-b2f14d9f44a39df2","name":"Triangle of coefficients of Legendre polynomials 2^n P_n (x).","first_index":0,"fingerprint":"f6-a866a9bf376c404a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["6","2"]],"card_rows":[["1"],["2"],["6","2"],["20","12"]],"row_sums_preview":["1","2","8","32","136","592","2624","11776"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A008556"],"derivation":{"root":"n-b2f14d9f44a39df2","steps":[],"edges":[],"text":"A008556"},"headline":"A008556: Triangle of coefficients of Legendre polynomials 2^n P_n (x).","chain":null}},{"id":"l-5815e6be9d2bab65","kind":"oeis_match","node_a_id":"n-eacb751f39dbd056","node_b_id":null,"fingerprint":null,"oeis_id":"A115951","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A115951","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ae21aef4c56f363b","kind":"oeis_match","node_a_id":"n-eacb751f39dbd056","node_b_id":null,"fingerprint":null,"oeis_id":"A006139","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":"A006139","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ffca7ebbd0168555","kind":"oeis_match","node_a_id":"n-eacb751f39dbd056","node_b_id":null,"fingerprint":null,"oeis_id":"A008556","in_catalog":1,"target":"reversed_triangle","row_alignment":"node_lengths","rows_matched":12,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":true,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":12,"target":"reversed_triangle","terms_compared":42},"in_catalog":true,"oeis_id":"A008556","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A115951","relation":"identification"},{"oeis_id":"A008556","relation":"match_reversed_triangle"},{"oeis_id":"A006139","relation":"match_row_sums"},{"oeis_id":"A115951","relation":"match_triangle"}],"bfiles":[{"node_id":"n-eacb751f39dbd056","oeis_id":"A006139","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":14357},{"node_id":"n-eacb751f39dbd056","oeis_id":"A008556","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":62,"terms_matched":42,"oeis_terms_agreeing":42,"oeis_terms_available":42,"term_count":992,"raw_size":27788},{"node_id":"n-eacb751f39dbd056","oeis_id":"A115951","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":13445}],"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: 0, 2;\n  2: 0, 2,  6;\n  3: 0, 0, 12, 20;\n  4: 0, 0,  6, 60,  70;\n  5: 0, 0,  0, 60, 280,  252;\n  6: 0, 0,  0, 20, 420, 1260,   924;\n  7: 0, 0,  0,  0, 280, 2520,  5544,  3432;\n  8: 0, 0,  0,  0,  70, 2520, 13860, 24024,  12870;\n  9: 0, 0,  0,  0,   0, 1260, 18480, 72072, 102960, 48620;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (4*n-2)*x*P(n-1,x) + (4*n-4)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 2)*x*P[n - 1, x] + (4*n - 4)*x*P[n - 2, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 2)*x*v[i - 1] + (4*n - 4)*x*v[i - 2])/(n)); 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":"A008556","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     1;\n  1:     2;\n  2:     6,      2;\n  3:    20,     12;\n  4:    70,     60,     6;\n  5:   252,    280,    60;\n  6:   924,   1260,   420,    20;\n  7:  3432,   5544,  2520,   280;\n  8: 12870,  24024, 13860,  2520,   70;\n  9: 48620, 102960, 72072, 18480, 1260;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where n*P(n,x) = (4*n-2)*x*P(n-1,x) + (4*n-4)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 2)*x*P[n - 1, x] + (4*n - 4)*x*P[n - 2, x])/(n)]];\nTable[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 2)*x*v[i - 1] + (4*n - 4)*x*v[i - 2])/(n)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":"low","regular_triangle":false}},{"oeis_id":"A115951","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: 0, 2;\n  2: 0, 2,  6;\n  3: 0, 0, 12, 20;\n  4: 0, 0,  6, 60,  70;\n  5: 0, 0,  0, 60, 280,  252;\n  6: 0, 0,  0, 20, 420, 1260,   924;\n  7: 0, 0,  0,  0, 280, 2520,  5544,  3432;\n  8: 0, 0,  0,  0,  70, 2520, 13860, 24024,  12870;\n  9: 0, 0,  0,  0,   0, 1260, 18480, 72072, 102960, 48620;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (4*n-2)*x*P(n-1,x) + (4*n-4)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 2)*x*P[n - 1, x] + (4*n - 4)*x*P[n - 2, x])/(n)]];\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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 2)*x*v[i - 1] + (4*n - 4)*x*v[i - 2])/(n)); 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-eacb751f39dbd056","steps":[],"edges":[],"text":"A115951"},"headline":"A115951: Expansion of 1/sqrt(1-4*x*y-4*x^2*y).","chain":null}