{"id":"n-d44f94dfc4c66d6b","revision":0,"first_index":0,"fingerprint":"f6-2f2b2d9b62c79a93","name":"reversal of basis_map_A013609 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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"],["2","2"],["8","14","6"],["48","116","92","24"],["384","1168","1316","652","120"],["3840","13984","20168","14416","5112","720"],["46080","194688","339904","314168","162256","44424","5040"],["645120","3094272","6316160","7117584","4784928","1919984","425952","40320"],["10321920","55314432","128907008","170726784","140617104","73784096","24095088","4478688","362880"],["185794560","1098878976","2873470464","4362152192","4238375712","2734284768","1171552544","321567264","51318720","3628800"],["3715891200","24021319680","69557078016","118851218944","132751188352","101307828192","53508183328","19318423264","4563614304","637081920","39916800"],["81749606400","573059727360","1818511552512","3449411752960","4346740771072","3821786480448","2392873971520","1067103511744","332220593856","68779173888","8523152640","479001600"],["1961990553600","14816178339840","51094053715968","106426532253696","149164131294208","148230505554688","107112199562304","56717845911616","21845639905216","5969567893440","1098684923904","122297022720","6227020800"],["51011754393600","412688504586240","1535871893372928","3482406590619648","5368238865201152","5942290982540800","4860144266385536","2974234787574272","1362036480298240","461047723902464","112139758529664","18561311525376","1874060858880","87178291200"],["1428329123020800","12320454444318720","49194740583235584","120545462937944064","202546787084926976","246907730489159680","225218404196907008","156180738047862656","82750543261964800","33339883473742592","10055629097367552","2201813100655488","330893376929280","30551905036800","1307674368000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","28","280","3640","58240","1106560","24344320","608608000","17041024000","528271744000","17961239296000","664565853952000","26582634158080000","1143053268797440000"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187152,"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-32708e5e8789204c"]},{"id":187153,"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-32708e5e8789204c"]},{"id":187154,"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-32708e5e8789204c"]}],"region_exact":{},"incoming_edges":[{"id":"e-32708e5e8789204c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-d44f94dfc4c66d6b","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"bb2deb78107ba0493072225cd36fbd34634fb3feb72653fa5c2e65e63c038b10","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-656046abebe1865b","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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-45175f26f5cf150e","kind":"fingerprint_related","node_a_id":"n-d44f94dfc4c66d6b","node_b_id":"n-bae2225dfab4d71f","fingerprint":"f6-2f2b2d9b62c79a93","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f2b2d9b62c79a93"},"other_id":"n-bae2225dfab4d71f","other":{"id":"n-bae2225dfab4d71f","name":"basis_map_A013609 of reversal of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-2f2b2d9b62c79a93","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["2","2"]],"card_rows":[["1"],["1"],["2","2"],["6","14","8"]],"row_sums_preview":["1","1","4","28","280","3640","58240","1106560"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["reverse","basis A013609"],"edges":["e-c7203dc00a9aba0b","e-bf209236cd0594b7"],"text":"A132393 \u2192 reverse \u2192 basis A013609"},"headline":"A132393 \u2192 reverse \u2192 basis A013609","chain":null}},{"id":"l-54a2b0dc07babd04","kind":"fingerprint_related","node_a_id":"n-656046abebe1865b","node_b_id":"n-d44f94dfc4c66d6b","fingerprint":"f6-2f2b2d9b62c79a93","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f2b2d9b62c79a93"},"other_id":"n-656046abebe1865b","other":{"id":"n-656046abebe1865b","name":"basis_map_A013609 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","first_index":0,"fingerprint":"f6-2f2b2d9b62c79a93","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["6","14","8"]],"card_rows":[["1"],["2","2"],["6","14","8"],["24","92","116","48"]],"row_sums_preview":["1","4","28","280","3640","58240","1106560","24344320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4d60c3dd8627c01d","steps":["basis A013609"],"edges":["e-23134a27b4e4bcc8"],"text":"A094638 \u2192 basis A013609"},"headline":"A094638 \u2192 basis A013609","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:         2,          2;\n  2:         8,         14,          6;\n  3:        48,        116,         92,         24;\n  4:       384,       1168,       1316,        652,        120;\n  5:      3840,      13984,      20168,      14416,       5112,        720;\n  6:     46080,     194688,     339904,     314168,     162256,      44424,       5040;\n  7:    645120,    3094272,    6316160,    7117584,    4784928,    1919984,     425952,     40320;\n  8:  10321920,   55314432,  128907008,  170726784,  140617104,   73784096,   24095088,   4478688,   362880;\n  9: 185794560, 1098878976, 2873470464, 4362152192, 4238375712, 2734284768, 1171552544, 321567264, 51318720, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+(n+1)*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[(2*n + (n + 1)*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] = (2*n + (n + 1)*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-4d60c3dd8627c01d","steps":["basis A013609","reverse"],"edges":["e-23134a27b4e4bcc8","e-32708e5e8789204c"],"text":"A094638 \u2192 basis A013609 \u2192 reverse"},"headline":"A094638 \u2192 basis A013609 \u2192 reverse","chain":null}