{"id":"n-8cf5a16a6dfee934","revision":1,"first_index":0,"fingerprint":"f6-da8d2afb3d35ceef","name":"Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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"],["1","4","2"],["1","13","18","6"],["1","40","116","96","24"],["1","121","660","1020","600","120"],["1","364","3542","9120","9480","4320","720"],["1","1093","18438","74466","121800","94920","35280","5040"],["1","3280","94376","576576","1394064","1653120","1028160","322560","40320"],["1","9841","478440","4319160","14852880","25154640","23284800","12035520","3265920","362880"],["1","29524","2411882","31669440","150952560","350965440","453630240","343526400","151804800","36288000","3628800"],["1","88573","12118458","228921726","1485250800","4615382640","8002985760","8328307680","5328892800","2055715200","439084800","39916800"],["1","265720","60769436","1638807456","14282944104","58195463040","131731110720","180945515520","157217639040","87018624000","29777932800","5748019200","479001600"],["1","797161","304378620","11653960500","135101726760","711564813960","2061677217600","3636300507840","4120263987840","3068312607360","1495522828800","459761702400","80951270400","6227020800"],["1","2391484","1523487422","82490859360","1262531382840","8502721587360","31071192712560","68976248140800","99134891856000","95380315430400","62089105478400","27025270272000","7540922188800","1220496076800","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","7","38","277","2522","27547","351038","5112457","83764082","1524907087","30536665238","667096092637","15787642820042","402374890155427"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A225476","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159670,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-2, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0)","region_exact":null,"edges":["e-0759b614e8dc4111"]},{"id":159671,"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-0759b614e8dc4111"]},{"id":159672,"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-0759b614e8dc4111"]},{"id":159673,"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-0759b614e8dc4111"]},{"id":159674,"position":4,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"[-1, 1]","output_interval":"[-2, 0]","statement":"source [-1, 1], target [-2, 0]"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in [-1, 1] to roots in [-2, 0]: source [-1, 1], target [-2, 0]","region_exact":null,"edges":["e-1a33bcef7c8475f7"]}],"region_exact":{},"incoming_edges":[{"id":"e-1a33bcef7c8475f7","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-8cf5a16a6dfee934","premise_count":0,"theorem":"interval basis maps: for L f = (a + b t) f + kappa (t - l)(t - h) f', l < h, kappa > 0, b + kappa deg g > 0, and g with positive leading coefficient, real-rooted in [l, h] with endpoint multiplicities m_l, m_h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in [l, h] if and only if S lies in [a + b l - kappa (h - l) m_l, a + b h + kappa (h - l) m_h] (RealRooted intervalBasisMap_maps_realRootedIn_iff; Ma and Wang, EJC 15 (2008) R17, Theorem 2 and Corollary 3); a negative leading coefficient of g only changes the sign of T","lean_route":null,"lean_declarations":[],"certificate_hash":"99050cf856d643310d31fa0bcac65a6bca8c95e56ba16ca0e56f1b2e47ad4fd5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"[-1, 1]","output_interval":"[-2, 0]","statement":"source [-1, 1], target [-2, 0]"}],"premises":[]},{"id":"e-0759b614e8dc4111","operation":{"operation":"theorem_check","theorem":"ma_wang_interval"},"operation_detail":"ma_wang_interval","conclusion_id":"n-8cf5a16a6dfee934","premise_count":0,"theorem":"S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction","lean_route":null,"lean_declarations":["RealRooted.affineEndpointStep_closed","RealRooted.affineEndpointStep_roots_mem_Icc_iff","RealRooted.affineEndpointStep_rootMultiplicity_zero","RealRooted.affineEndpointStep_rootMultiplicity_neg_one","RealRooted.MaWang.endpointStep","RealRooted.MaWang.affineEndpointSequence","RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff"],"certificate_hash":"f6c0abf15546d5f528171ee7769c2ee28c7e7c5a272a9a5c702b08e6a9c409b8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-b0c8196dbc01af81","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-d519174be8286cb2","conclusion_name":"obreschkoff_combination of Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0."},{"id":"e-b0c8196dbc01af81","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-d519174be8286cb2","conclusion_name":"obreschkoff_combination of Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0."}],"used_by_total":2,"links":[{"id":"l-51bc29faa581a093","kind":"oeis_match","node_a_id":"n-8cf5a16a6dfee934","node_b_id":null,"fingerprint":null,"oeis_id":"A337026","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":"A337026","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d4aed179660c8073","kind":"oeis_match","node_a_id":"n-8cf5a16a6dfee934","node_b_id":null,"fingerprint":null,"oeis_id":"A225476","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":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":49},"in_catalog":true,"oeis_id":"A225476","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A225476","relation":"identification"},{"oeis_id":"A337026","relation":"match_row_sums"},{"oeis_id":"A225476","relation":"match_triangle"}],"bfiles":[{"node_id":"n-8cf5a16a6dfee934","oeis_id":"A225476","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":49,"oeis_terms_agreeing":49,"oeis_terms_available":49,"term_count":990,"raw_size":37560},{"node_id":"n-8cf5a16a6dfee934","oeis_id":"A337026","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":20,"oeis_terms_available":20,"term_count":200,"raw_size":39245}],"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,    4,      2;\n  3: 1,   13,     18,       6;\n  4: 1,   40,    116,      96,       24;\n  5: 1,  121,    660,    1020,      600,      120;\n  6: 1,  364,   3542,    9120,     9480,     4320,      720;\n  7: 1, 1093,  18438,   74466,   121800,    94920,    35280,     5040;\n  8: 1, 3280,  94376,  576576,  1394064,  1653120,  1028160,   322560,   40320;\n  9: 1, 9841, 478440, 4319160, 14852880, 25154640, 23284800, 12035520, 3265920, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (2*x+x^2)*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[(1 + x)*P[n - 1, x] + (2*x + x^2)*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] = (1 + x)*v[i - 1] + (2*x + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A225476","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: 1,    4,      2;\n  3: 1,   13,     18,       6;\n  4: 1,   40,    116,      96,       24;\n  5: 1,  121,    660,    1020,      600,      120;\n  6: 1,  364,   3542,    9120,     9480,     4320,      720;\n  7: 1, 1093,  18438,   74466,   121800,    94920,    35280,     5040;\n  8: 1, 3280,  94376,  576576,  1394064,  1653120,  1028160,   322560,   40320;\n  9: 1, 9841, 478440, 4319160, 14852880, 25154640, 23284800, 12035520, 3265920, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (2*x+x^2)*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[(1 + x)*P[n - 1, x] + (2*x + x^2)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (1 + x)*v[i - 1] + (2*x + x^2)*deriv(v[i - 1], x)); 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-8cf5a16a6dfee934","steps":[],"edges":[],"text":"A225476"},"headline":"A225476: Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","chain":null}