{"id":"n-426fc75b9e92ccc7","revision":0,"first_index":0,"fingerprint":"f6-87d1d1ae6917d3b6","name":"mobius_x_over_x_plus_one of one_plus_derivative of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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"],["1","2"],["1","5","5"],["2","14","28","17"],["6","52","149","176","74"],["24","244","891","1524","1246","394"],["120","1388","6089","13441","16029","9892","2484"],["720","9288","47244","126664","196386","177715","87424","18108"],["5040","71496","411816","1291928","2448529","2894928","2096464","853400","149904"],["40320","622368","3991380","14287496","31857813","46239150","43904176","26388348","9133128","1389456"],["362880","6044832","42618564","170947580","436944661","748854261","875540665","691783676","354344748","106438608","14257440"],["3628800","64802880","497334168","2205021488","6340477606","12495541750","17296145560","16866120941","11379903800","5068229484","1342687248","160460640"],["39916800","760006080","6298855848","30541870360","97405744006","216539452988","345444209109","399591208160","333331457458","195865076408","77047758072","18235771776","1965444480"],["479001600","9678908160","86065894656","452498979360","1582939287788","3912181284868","7054017724183","9420361941704","9336134476670","6791911407566","3529541457872","1241906486040","265390944768","26029779840"],["6227020800","133010830080","1262074808448","7144540255776","27170957309924","73810043440992","148213491615045","224052632323521","256700876183465","222138516150552","143082347139016","66579231946384","21172592828880","4121235896832","370643938560"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","11","61","457","4323","49443","663549","10223505","177853635","3448137915","73720354365","1723066771545","43709137575075","1195958421688275"],"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"],["1","2"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["-4","-12","-13","-6","-1"],["4","14","18","10","2"],["-1","-4","-6","-4","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","-7","-5","-1"],["2","6","6","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176861,"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-d61666c9a8af3460"]},{"id":176862,"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-d61666c9a8af3460"]},{"id":176863,"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-d61666c9a8af3460"]}],"region_exact":{},"incoming_edges":[{"id":"e-d61666c9a8af3460","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-426fc75b9e92ccc7","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":"ff29be8845f0b5c1e66ebbd4c58897c671b29e533268553791d6126967c380af","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-125f84bee6f03b65","role":"sequence","name":"one_plus_derivative of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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":[],"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:     1,      2;\n  2:     1,      5,       5;\n  3:     2,     14,      28,       17;\n  4:     6,     52,     149,      176,       74;\n  5:    24,    244,     891,     1524,     1246,      394;\n  6:   120,   1388,    6089,    13441,    16029,     9892,     2484;\n  7:   720,   9288,   47244,   126664,   196386,   177715,    87424,    18108;\n  8:  5040,  71496,  411816,  1291928,  2448529,  2894928,  2096464,   853400,  149904;\n  9: 40320, 622368, 3991380, 14287496, 31857813, 46239150, 43904176, 26388348, 9133128, 1389456;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(n^2-4*n+4+(4*n^2-14*n+12)*x+(6*n^2-18*n+13)*x^2+(4*n^2-10*n+6)*x^3+(n^2-2*n+1)*x^4)*P(n-2,x) + (2*n-3+(6*n-7)*x+(6*n-5)*x^2+(2*n-1)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 4*n + 4 + (4*n^2 - 14*n + 12)*x + (6*n^2 - 18*n + 13)*x^2 + (4*n^2 - 10*n + 6)*x^3 + (n^2 - 2*n + 1)*x^4)*P[n - 2, x] + (2*n - 3 + (6*n - 7)*x + (6*n - 5)*x^2 + (2*n - 1)*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 4*n + 4 + (4*n^2 - 14*n + 12)*x + (6*n^2 - 18*n + 13)*x^2 + (4*n^2 - 10*n + 6)*x^3 + (n^2 - 2*n + 1)*x^4)*v[i - 2] + (2*n - 3 + (6*n - 7)*x + (6*n - 5)*x^2 + (2*n - 1)*x^3)*v[i - 1])/(1 + 2*x + x^2)); 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-b9fbfb57d9269e2f","steps":["D+1","x \u21a6 x/(x+1)"],"edges":["e-649ce40fa3feb4c2","e-d61666c9a8af3460"],"text":"A132393 \u2192 D+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A132393 \u2192 D+1 \u2192 x \u21a6 x/(x+1)","chain":null}
