{"id":"n-0d0c19e5a4636085","revision":0,"first_index":0,"fingerprint":"f6-7d8c365492dbd6da","name":"reversal of theta_plus_one 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":false,"rows":[["1"],["2","1"],["6","6","1"],["24","33","12","1"],["120","200","105","20","1"],["720","1370","900","255","30","1"],["5040","10584","8120","2940","525","42","1"],["40320","91476","78792","33845","7840","966","56","1"],["362880","876672","826868","403704","112245","18144","1638","72","1"],["3628800","9239184","9381600","5065760","1615950","316365","37800","2610","90","1"],["39916800","106286400","114782184","67276000","23918510","5412330","788865","72600","3960","110","1"],["479001600","1325982240","1509179760","947322684","367965840","93376745","15825348","1787115","130680","5775","132","1"],["6227020800","17837314560","21247155072","14140148880","5914861524","1648561200","314931617","41559804","3747315","223080","8151","156","1"],["87178291200","257435867520","319160604672","223451284056","99577037560","30025069074","6327545224","945255311","100017918","7372365","364364","11193","182","1"],["1307674368000","3968519063040","5098038371712","3731871124800","1756937662480","566633667600","129683906352","21451630200","2578881305","223873650","13748735","573300","15015","210","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","13","70","446","3276","27252","253296","2602224","29288160","358457760","4740577920","67375532160","1024208720640","16583626886400"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["-1","2","-1"],["2","-2","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1"],["-2","2"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181332,"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-6411ace3ea682158"]},{"id":181333,"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-6411ace3ea682158"]},{"id":181334,"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-6411ace3ea682158"]}],"region_exact":{},"incoming_edges":[{"id":"e-6411ace3ea682158","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-0d0c19e5a4636085","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":"fbd59b79c6a36a28e726221ffade298ba8716c2661380d35fb632ace7a99a6e5","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-994dbf40e5f5e735","role":"sequence","name":"theta_plus_one 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-0c155f7a69efe841","kind":"fingerprint_related","node_a_id":"n-994dbf40e5f5e735","node_b_id":"n-0d0c19e5a4636085","fingerprint":"f6-7d8c365492dbd6da","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7d8c365492dbd6da"},"other_id":"n-994dbf40e5f5e735","other":{"id":"n-994dbf40e5f5e735","name":"theta_plus_one 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":"f6-7d8c365492dbd6da","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","12","33","24"]],"row_sums_preview":["1","3","13","70","446","3276","27252","253296"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4d60c3dd8627c01d","steps":["\u03b8+1"],"edges":["e-8baeeca6d59e95f0"],"text":"A094638 \u2192 \u03b8+1"},"headline":"A094638 \u2192 \u03b8+1","chain":null}},{"id":"l-dc74b7695f0e9646","kind":"fingerprint_related","node_a_id":"n-ed516dafc23c3127","node_b_id":"n-0d0c19e5a4636085","fingerprint":"f6-7d8c365492dbd6da","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7d8c365492dbd6da"},"other_id":"n-ed516dafc23c3127","other":{"id":"n-ed516dafc23c3127","name":"theta_plus_one 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":"f6-7d8c365492dbd6da","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","2"]],"card_rows":[["1"],["1"],["1","2"],["1","6","6"]],"row_sums_preview":["1","1","3","13","70","446","3276","27252"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["reverse","\u03b8+1"],"edges":["e-c7203dc00a9aba0b","e-75b245d54c2ea25e"],"text":"A132393 \u2192 reverse \u2192 \u03b8+1"},"headline":"A132393 \u2192 reverse \u2192 \u03b8+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:       2,       1;\n  2:       6,       6,       1;\n  3:      24,      33,      12,       1;\n  4:     120,     200,     105,      20,       1;\n  5:     720,    1370,     900,     255,      30,      1;\n  6:    5040,   10584,    8120,    2940,     525,     42,     1;\n  7:   40320,   91476,   78792,   33845,    7840,    966,    56,    1;\n  8:  362880,  876672,  826868,  403704,  112245,  18144,  1638,   72,  1;\n  9: 3628800, 9239184, 9381600, 5065760, 1615950, 316365, 37800, 2610, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3-2*n^2+n+(2*n^2-2*n)*x+n*x^2)*P(n-2,x) + (2*n^2-2*n+(2*n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 2*n^2 + n + (2*n^2 - 2*n)*x + n*x^2)*P[n - 2, x] + (2*n^2 - 2*n + (2*n - 1)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 2*n^2 + n + (2*n^2 - 2*n)*x + n*x^2)*v[i - 2] + (2*n^2 - 2*n + (2*n - 1)*x)*v[i - 1])/(n - 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":["\u03b8+1","reverse"],"edges":["e-8baeeca6d59e95f0","e-6411ace3ea682158"],"text":"A094638 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A094638 \u2192 \u03b8+1 \u2192 reverse","chain":null}