{"id":"n-fb9493e648020cca","revision":1,"first_index":0,"fingerprint":"f6-3f80906410b0a92d","name":"basis_map_A013609 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","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":true,"rows":[["1"],["3","2"],["12","14","4"],["60","94","48","8"],["360","684","476","144","16"],["2520","5508","4700","1960","400","32"],["20160","49104","48616","25080","7120","1056","64"],["181440","482256","535752","322952","114240","23744","2688","128"],["1814400","5185440","6322032","4301024","1788304","465920","74368","6656","256"],["19958400","60668640","79913232","59955328","28273392","8701728","1749888","221952","16128","512"],["239500800","767940480","1080296064","879290400","459191360","160967520","38402112","6163200","637440","38400","1024"],["3113510400","10462227840","15579729792","13591367328","7728068480","3010960480","821162496","156925824","20613120","1774080","90112","2048"],["43589145600","152698210560","239040672768","221438602176","135375693376","57609583680","17518195904","3839286528","602435328","66063360","4809728","208896","4096"],["653837184000","2377651449600","3891006512640","3799660378176","2473512604992","1134895141952","377992105920","92625689728","16715102976","2195821056","204272640","12752896","479232","8192"],["10461394944000","39350097561600","67011407101440","68576579076096","47175522436224","23105347481216","8317663978624","2237995247488","452693027072","68563342848","7660004352","612591616","33173504","1089536","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","30","210","1680","15120","151200","1663200","19958400","259459200","3632428800","54486432000","871782912000","14820309504000","266765571072000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166709,"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-3581c2f82a666ba6"]},{"id":166710,"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-3581c2f82a666ba6"]},{"id":166711,"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-3581c2f82a666ba6"]}],"region_exact":{},"incoming_edges":[{"id":"e-3581c2f82a666ba6","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-fb9493e648020cca","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"a87dfe7a895a97c24f21c77bcc5f5ba9665d22809947f1dfa82684b06bb53e91","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-914700b1060feeb2","role":"sequence","name":"Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-7b4b119d8a63eeb2","kind":"fingerprint_related","node_a_id":"n-fb9493e648020cca","node_b_id":"n-578e4bbd7f77169d","fingerprint":"f6-3f80906410b0a92d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3f80906410b0a92d"},"other_id":"n-578e4bbd7f77169d","other":{"id":"n-578e4bbd7f77169d","name":"basis_map_A013609 of Unsigned 2-Stirling numbers of the first kind.","first_index":0,"fingerprint":"f6-3f80906410b0a92d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["12","14","4"],["60","94","48","8"],["360","684","476","144","16"]],"card_rows":[["12","14","4"],["60","94","48","8"],["360","684","476","144","16"],["2520","5508","4700","1960","400","32"]],"row_sums_preview":["30","210","1680","15120","151200","1663200","19958400","259459200"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c237629cf5fdbb07","steps":["basis A013609"],"edges":["e-7cc986a9eefcf6a5"],"text":"A143491 \u2192 basis A013609"},"headline":"A143491 \u2192 basis A013609","chain":null}},{"id":"l-04347968bb6e3c7a","kind":"oeis_match","node_a_id":"n-fb9493e648020cca","node_b_id":null,"fingerprint":null,"oeis_id":"A001720","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":4,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001720","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001720","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-fb9493e648020cca","oeis_id":"A001720","target":"row_sums","oeis_offset":4,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":20,"oeis_terms_available":20,"term_count":200,"raw_size":35307}],"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:        3,        2;\n  2:       12,       14,        4;\n  3:       60,       94,       48,        8;\n  4:      360,      684,      476,      144,       16;\n  5:     2520,     5508,     4700,     1960,      400,      32;\n  6:    20160,    49104,    48616,    25080,     7120,    1056,      64;\n  7:   181440,   482256,   535752,   322952,   114240,   23744,    2688,    128;\n  8:  1814400,  5185440,  6322032,  4301024,  1788304,  465920,   74368,   6656,   256;\n  9: 19958400, 60668640, 79913232, 59955328, 28273392, 8701728, 1749888, 221952, 16128, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+2+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n + 2 + 2*x)*P[n - 1, 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] = (n + 2 + 2*x)*v[i - 1]); 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-914700b1060feeb2","steps":["basis A013609"],"edges":["e-3581c2f82a666ba6"],"text":"A136124 \u2192 basis A013609"},"headline":"A136124 \u2192 basis A013609","chain":null}
