{"id":"n-a7e9ebb9860a9dbe","revision":1,"first_index":0,"fingerprint":"f6-7dbaacaea3b384bc","name":"Stirling2 triangle with scaled diagonals (powers of 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"],["2","1"],["4","6","1"],["8","28","12","1"],["16","120","100","20","1"],["32","496","720","260","30","1"],["64","2016","4816","2800","560","42","1"],["128","8128","30912","27216","8400","1064","56","1"],["256","32640","193600","248640","111216","21168","1848","72","1"],["512","130816","1194240","2182720","1360800","365232","47040","3000","90","1"],["1024","523776","7296256","18656000","15790720","5743584","1023792","95040","4620","110","1"],["2048","2096128","44301312","156544256","176563200","84713728","20076672","2544432","178200","6820","132","1"],["4096","8386560","267904000","1296655360","1922176256","1193127936","365787136","60787584","5752032","314600","9724","156","1"],["8192","33550336","1615810560","10641146880","20518417920","16239711488","6314147840","1338388480","164324160","12044032","528528","13468","182","1"],["16384","134209536","9728413696","86744985600","215825326080","215394955776","104637781248","27728363520","4296223360","405204800","23671648","851760","18200","210","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","11","49","257","1539","10299","75905","609441","5284451","49134923","487026929","5120905441","56878092067","664920021819"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A075497","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158817,"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-ca5f07049cdde3bf"]},{"id":158818,"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-ca5f07049cdde3bf"]},{"id":158819,"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-ca5f07049cdde3bf"]},{"id":158820,"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-ca5f07049cdde3bf"]},{"id":158821,"position":4,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 2]","output_interval":"(-inf, 0]","statement":"source (-inf, a + b h + kappa m] = (-inf, 2] with m = 0, target (-inf, 0]"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 2] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 2] with m = 0, target (-inf, 0]","region_exact":null,"edges":["e-16c10c145478dde5"]}],"region_exact":{},"incoming_edges":[{"id":"e-16c10c145478dde5","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-a7e9ebb9860a9dbe","premise_count":0,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"7cefbc9e40378358015add61947505e866ca468d95f9c975b044e578e409f855","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 2]","output_interval":"(-inf, 0]","statement":"source (-inf, a + b h + kappa m] = (-inf, 2] with m = 0, target (-inf, 0]"}],"premises":[]},{"id":"e-ca5f07049cdde3bf","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-a7e9ebb9860a9dbe","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":"ed87dba519393e5f8e63d196c6b5d967dafaabe0445132abc7cc57e925742ef2","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-b1f98f91a12826e9","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-b4652f0ae8d5ecf9","conclusion_name":"obreschkoff_combination of Stirling2 triangle with scaled diagonals (powers of 2)."},{"id":"e-b1f98f91a12826e9","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-b4652f0ae8d5ecf9","conclusion_name":"obreschkoff_combination of Stirling2 triangle with scaled diagonals (powers of 2)."},{"id":"e-188635230e4781f5","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-7e0e90f00bfc1e53","conclusion_name":"reversal of Stirling2 triangle with scaled diagonals (powers of 2)."},{"id":"e-3e5da36e28dd2977","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-a3fdc04d216a33db","conclusion_name":"theta_plus_one of Stirling2 triangle with scaled diagonals (powers of 2)."},{"id":"e-6dbd3471b7b80662","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-7eb19ab4ebf03277","conclusion_name":"shift_x_plus_one of Stirling2 triangle with scaled diagonals (powers of 2)."},{"id":"e-775d8a1aaaf30263","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-28f29d8563043b26","conclusion_name":"derivative of Stirling2 triangle with scaled diagonals (powers of 2)."},{"id":"e-8941cbe9a26788e8","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-276e0e2ea4eaffea","conclusion_name":"mobius_x_over_x_plus_one of Stirling2 triangle with scaled diagonals (powers of 2)."},{"id":"e-9fe96d700c9448ac","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-df88941ef8650983","conclusion_name":"one_plus_derivative of Stirling2 triangle with scaled diagonals (powers of 2)."},{"id":"e-d445ae4d7e7f6688","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-cc6a74b7fca20c9d","conclusion_name":"theta_plus_one_squared of Stirling2 triangle with scaled diagonals (powers of 2)."}],"used_by_total":9,"links":[{"id":"l-732f2103676739f4","kind":"fingerprint_related","node_a_id":"n-a7e9ebb9860a9dbe","node_b_id":"n-7e0e90f00bfc1e53","fingerprint":"f6-7dbaacaea3b384bc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7dbaacaea3b384bc"},"other_id":"n-7e0e90f00bfc1e53","other":{"id":"n-7e0e90f00bfc1e53","name":"reversal of Stirling2 triangle with scaled diagonals (powers of 2).","first_index":0,"fingerprint":"f6-7dbaacaea3b384bc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","4"]],"card_rows":[["1"],["1","2"],["1","6","4"],["1","12","28","8"]],"row_sums_preview":["1","3","11","49","257","1539","10299","75905"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a7e9ebb9860a9dbe","steps":["reverse"],"edges":["e-188635230e4781f5"],"text":"A075497 \u2192 reverse"},"headline":"A075497 \u2192 reverse","chain":null}},{"id":"l-d69698a22ea75f24","kind":"oeis_match","node_a_id":"n-a7e9ebb9860a9dbe","node_b_id":null,"fingerprint":null,"oeis_id":"A075497","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":"A075497","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e4e37978054508cf","kind":"oeis_match","node_a_id":"n-a7e9ebb9860a9dbe","node_b_id":null,"fingerprint":null,"oeis_id":"A004211","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A004211","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A075497","relation":"identification"},{"oeis_id":"A004211","relation":"match_row_sums"},{"oeis_id":"A075497","relation":"match_triangle"}],"bfiles":[{"node_id":"n-a7e9ebb9860a9dbe","oeis_id":"A075497","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":25495}],"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:   2,      1;\n  2:   4,      6,       1;\n  3:   8,     28,      12,       1;\n  4:  16,    120,     100,      20,       1;\n  5:  32,    496,     720,     260,      30,      1;\n  6:  64,   2016,    4816,    2800,     560,     42,     1;\n  7: 128,   8128,   30912,   27216,    8400,   1064,    56,    1;\n  8: 256,  32640,  193600,  248640,  111216,  21168,  1848,   72,  1;\n  9: 512, 130816, 1194240, 2182720, 1360800, 365232, 47040, 3000, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + 2*x*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[(2 + x)*P[n - 1, x] + 2*x*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] = (2 + x)*v[i - 1] + 2*x*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":"A075497","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:   2,      1;\n   3:   4,      6,       1;\n   4:   8,     28,      12,       1;\n   5:  16,    120,     100,      20,       1;\n   6:  32,    496,     720,     260,      30,      1;\n   7:  64,   2016,    4816,    2800,     560,     42,     1;\n   8: 128,   8128,   30912,   27216,    8400,   1064,    56,    1;\n   9: 256,  32640,  193600,  248640,  111216,  21168,  1848,   72,  1;\n  10: 512, 130816, 1194240, 2182720, 1360800, 365232, 47040, 3000, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + 2*x*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[(2 + x)*P[n - 1, x] + 2*x*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] = (2 + x)*v[i - 1] + 2*x*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":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-a7e9ebb9860a9dbe","steps":[],"edges":[],"text":"A075497"},"headline":"A075497: Stirling2 triangle with scaled diagonals (powers of 2).","chain":null}
