{"id":"n-65034f179833a7f6","revision":1,"first_index":0,"fingerprint":"f6-68c7410893bd483f","name":"Stirling2 triangle with scaled diagonals (powers of 10).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["100","30","1"],["1000","700","60","1"],["10000","15000","2500","100","1"],["100000","310000","90000","6500","150","1"],["1000000","6300000","3010000","350000","14000","210","1"],["10000000","127000000","96600000","17010000","1050000","26600","280","1"],["100000000","2550000000","3025000000","777000000","69510000","2646000","46200","360","1"],["1000000000","51100000000","93300000000","34105000000","4252500000","228270000","5880000","75000","450","1"],["10000000000","1023000000000","2850100000000","1457500000000","246730000000","17948700000","639870000","11880000","115500","550","1"],["100000000000","20470000000000","86526000000000","61150100000000","13794000000000","1323652000000","62739600000","1590270000","22275000","170500","660","1"],["1000000000000","409500000000000","2616250000000000","2532530000000000","750850100000000","93213120000000","5715424000000","189961200000","3595020000","39325000","243100","780","1"],["10000000000000","8191000000000000","78897000000000000","103917450000000000","40075035000000000","6343637300000000","493292800000000","20912320000000","513513000000","7527520000","66066000","336700","910","1"],["100000000000000","163830000000000000","2375101000000000000","4235595000000000000","2107669200000000000","420693273000000000","40874133300000000","2166278400000000","67128490000000","1266265000000","14794780000","106470000","455000","1050","1"],["1000000000000000","3276700000000000000","71416860000000000000","171798901000000000000","109619055000000000000","27349265580000000000","3281882604000000000","214176405300000000","8207842500000000","193754990000000","2893690800000","27571180000","165620000","602000","1200","1"],["10000000000000000","65535000000000000000","2145782500000000000000","6943372900000000000000","5652751651000000000000","1750574989800000000000","257081047860000000000","20415995028000000000","952882230300000000","27583341500000000","512060978000000","6202232400000","49101780000","249900000","782000","1360","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["131","1761","27601","506651","10674211","251686881","6524202561","183991725451","5605930566051","183428104316161","6409252239788881","237948848526923611","9346097294356706051","386966245108218203201","16836505067572362863361"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A075505","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["100","30","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["10","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","10"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158857,"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-b1938fcbfe117d00"]},{"id":158858,"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-b1938fcbfe117d00"]},{"id":158859,"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-b1938fcbfe117d00"]},{"id":158860,"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-b1938fcbfe117d00"]},{"id":158861,"position":4,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 10]","output_interval":"(-inf, 0]","statement":"source (-inf, a + b h + kappa m] = (-inf, 10] with m = 0, target (-inf, 0]"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 10] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 10] with m = 0, target (-inf, 0]","region_exact":null,"edges":["e-506dd990cfc1b4f3"]}],"region_exact":{},"incoming_edges":[{"id":"e-506dd990cfc1b4f3","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-65034f179833a7f6","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":"cb84ae90f3ae1033302681f5626fe2c70bb18717e62edc3478209827a68f51e3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 10]","output_interval":"(-inf, 0]","statement":"source (-inf, a + b h + kappa m] = (-inf, 10] with m = 0, target (-inf, 0]"}],"premises":[]},{"id":"e-b1938fcbfe117d00","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-65034f179833a7f6","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":"92cdb3e357de676890ff7fe465448f55b46c3e76c8222d6f6d907c95fc051a6e","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-f660dc27e1b4f4f9","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-d745ee9fccac5a4d","conclusion_name":"obreschkoff_combination of Stirling2 triangle with scaled diagonals (powers of 10)."},{"id":"e-f660dc27e1b4f4f9","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-d745ee9fccac5a4d","conclusion_name":"obreschkoff_combination of Stirling2 triangle with scaled diagonals (powers of 10)."}],"used_by_total":2,"links":[{"id":"l-f9fd80b77cd9c6b9","kind":"oeis_match","node_a_id":"n-65034f179833a7f6","node_b_id":null,"fingerprint":null,"oeis_id":"A075509","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A075509","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A075509","relation":"match_row_sums"},{"oeis_id":"A075505","relation":"tail"}],"bfiles":[],"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:          100,             30,              1;\n  1:         1000,            700,             60,              1;\n  2:        10000,          15000,           2500,            100,              1;\n  3:       100000,         310000,          90000,           6500,            150,             1;\n  4:      1000000,        6300000,        3010000,         350000,          14000,           210,           1;\n  5:     10000000,      127000000,       96600000,       17010000,        1050000,         26600,         280,          1;\n  6:    100000000,     2550000000,     3025000000,      777000000,       69510000,       2646000,       46200,        360,        1;\n  7:   1000000000,    51100000000,    93300000000,    34105000000,     4252500000,     228270000,     5880000,      75000,      450,      1;\n  8:  10000000000,  1023000000000,  2850100000000,  1457500000000,   246730000000,   17948700000,   639870000,   11880000,   115500,    550,   1;\n  9: 100000000000, 20470000000000, 86526000000000, 61150100000000, 13794000000000, 1323652000000, 62739600000, 1590270000, 22275000, 170500, 660, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (10+x)*P(n-1,x) + 10*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 100 + 30*x + x^2.","mathematica":"P[0, x_] := 100 + 30*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(10 + x)*P[n - 1, x] + 10*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] = 100 + 30*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (10 + x)*v[i - 1] + 10*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}}],"derivation":{"root":"n-65034f179833a7f6","steps":[],"edges":[],"text":"A075505"},"headline":"Stirling2 triangle with scaled diagonals (powers of 10).","chain":null}