{"id":"n-462f527ec45d40b3","revision":0,"first_index":0,"fingerprint":"f6-f373be5c339f8575","name":"mobius_x_over_x_plus_one of 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","5"],["12","38","30"],["60","274","416","210"],["360","2124","4688","4588","1680"],["2520","18108","51932","74308","53052","15120"],["20160","170064","596536","1113784","1167496","651480","151200"],["181440","1752336","7239528","16585952","22759088","18705776","8527080","1663200"],["1814400","19700640","93423312","252733856","426622304","460166816","309740112","118956960","19958400"],["19958400","240294240","1283764752","3994575472","7978385472","10607924928","9389309840","5335148016","1765982880","259459200"],["239500800","3162948480","18769296384","65907612192","151664682272","238992495744","261182667072","195472113952","95883866784","27837270720","3632428800"],["3113510400","44710842240","291445080192","1138338404256","2960255052416","5381872678752","6980262108096","6458877487456","4178571977472","1800142521120","464780635200","54486432000"],["43589145600","675767957760","4795604598528","20599858942656","59656985201920","122710298341184","183833632373376","202108478553920","161842047483904","92059146935232","35309209230720","8199300211200","871782912000"],["653837184000","10877534841600","83422124259840","390523162314816","1245052380053952","2854823223550400","4843579557400768","6156798928656192","5863474847675200","4132202011254848","2094643636359744","723246060090240","152464847270400","14820309504000"],["10461394944000","185809626777600","1530549615306240","7749968833714176","26950255002529920","68088114417777536","128884090942319488","185693214891712896","204637978278614656","171657779438231168","107893934384343168","49275522415919232","15457866637950720","2981492202931200","266765571072000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","80","960","13440","215040","3870720","77414400","1703116800","40874803200","1062744883200","29756856729600","892705701888000","28566582460416000","971263803654144000"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["2","6","4"],["1","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183842,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-d227bf1cb6bf991b"]},{"id":183843,"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-d227bf1cb6bf991b"]},{"id":183844,"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-d227bf1cb6bf991b"]}],"region_exact":{},"incoming_edges":[{"id":"e-d227bf1cb6bf991b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-462f527ec45d40b3","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"5ca4899adb194e7a6a8e5ddcda7101ec33086550d628d9ef979c1c0cd13e70f1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-fb9493e648020cca","role":"sequence","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)).","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-2ce568571b58323f","kind":"fingerprint_related","node_a_id":"n-462f527ec45d40b3","node_b_id":"n-93bd186d4ea2fd62","fingerprint":"f6-f373be5c339f8575","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f373be5c339f8575"},"other_id":"n-93bd186d4ea2fd62","other":{"id":"n-93bd186d4ea2fd62","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Unsigned 2-Stirling numbers of the first kind.","first_index":0,"fingerprint":"f6-f373be5c339f8575","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["12","38","30"],["60","274","416","210"],["360","2124","4688","4588","1680"]],"card_rows":[["12","38","30"],["60","274","416","210"],["360","2124","4688","4588","1680"],["2520","18108","51932","74308","53052","15120"]],"row_sums_preview":["80","960","13440","215040","3870720","77414400","1703116800","40874803200"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c237629cf5fdbb07","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-7cc986a9eefcf6a5","e-797b961f7720edc1"],"text":"A143491 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A143491 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}}],"oeis":[],"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:        1;\n  1:        3,         5;\n  2:       12,        38,         30;\n  3:       60,       274,        416,        210;\n  4:      360,      2124,       4688,       4588,       1680;\n  5:     2520,     18108,      51932,      74308,      53052,       15120;\n  6:    20160,    170064,     596536,    1113784,    1167496,      651480,     151200;\n  7:   181440,   1752336,    7239528,   16585952,   22759088,    18705776,    8527080,    1663200;\n  8:  1814400,  19700640,   93423312,  252733856,  426622304,   460166816,  309740112,  118956960,   19958400;\n  9: 19958400, 240294240, 1283764752, 3994575472, 7978385472, 10607924928, 9389309840, 5335148016, 1765982880, 259459200;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n+2+(2*n+6)*x+(n+4)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 2 + (2*n + 6)*x + (n + 4)*x^2)*P[n - 1, x])/(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*n + 6)*x + (n + 4)*x^2)*v[i - 1])/(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-914700b1060feeb2","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-3581c2f82a666ba6","e-d227bf1cb6bf991b"],"text":"A136124 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A136124 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}