{"id":"n-772cd5e88801d525","revision":0,"first_index":1,"fingerprint":"f6-828e173002afaa4c","name":"mobius_x_over_x_plus_one of Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["0","1"],["0","2","3"],["0","5","15","11"],["0","16","71","103","49"],["0","65","376","803","752","261"],["0","326","2299","6406","8829","6026","1631"],["0","1957","16171","55134","99381","99974","53255","11743"],["0","13700","129275","518631","1147614","1513681","1190734","517511","95901"],["0","109601","1159880","5334398","13934096","22622179","23385328","15037350","5501032","876809"],["0","986410","11551011","59775612","179506130","344876526","439771569","372310828","201847914","63604938","8877691"],["0","9864101","126486303","726270033","2460009826","5445292425","8232931533","8612806271","6157407450","2879583279","795622007","98641011"],["0","108505112","1510830639","9520921895","35855544691","89686992858","156493672411","194414480261","171992959533","106205180258","43602691901","10713175023","1193556233"],["0","1302061345","19551510320","134043439441","554985589368","1545901129618","3052616853056","4382536077765","4609935730808","3526689527866","1913873362160","699444810569","154578737032","15624736141"],["0","16926797486","272517974299","2018067747554","9103551213327","27916371605552","61464474453238","99979692069258","121669354288195","110787422397224","74558497844830","36050842911370","11861103298823","2380223911586","220048367319"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","5","31","239","2257","25517","337615","5127047","87960673","1683108629","35544914239","821298510815","20611083565489","558299094880061"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["3","9","9","3"],["-1","-3","-3","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","-1","0"],["1","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177224,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-60865b531221d46e"]},{"id":177225,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-60865b531221d46e"]},{"id":177226,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-60865b531221d46e"]}],"region_exact":{},"incoming_edges":[{"id":"e-60865b531221d46e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-772cd5e88801d525","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":"49741e78c425d1a9b5fff4219c99633e71b62707f9f6616da05f30dcae6ee079","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cd3918419c5686a8","role":"sequence","name":"Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-9f82a18469afdee2","kind":"fingerprint_related","node_a_id":"n-772cd5e88801d525","node_b_id":"n-b271e0de26c0d825","fingerprint":"f6-828e173002afaa4c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-828e173002afaa4c"},"other_id":"n-b271e0de26c0d825","other":{"id":"n-b271e0de26c0d825","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","first_index":1,"fingerprint":"f6-828e173002afaa4c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["5","15","11"]],"card_rows":[["1"],["2","3"],["5","15","11"],["16","71","103","49"]],"row_sums_preview":["1","5","31","239","2257","25517","337615","5127047"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3272d5bc697b9cb4","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-77ff853dcb30433c","e-7ff55e3f9eb8c07f"],"text":"A094816 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A094816 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0,      1;\n   3: 0,      2,       3;\n   4: 0,      5,      15,      11;\n   5: 0,     16,      71,     103,       49;\n   6: 0,     65,     376,     803,      752,      261;\n   7: 0,    326,    2299,    6406,     8829,     6026,     1631;\n   8: 0,   1957,   16171,   55134,    99381,    99974,    53255,    11743;\n   9: 0,  13700,  129275,  518631,  1147614,  1513681,  1190734,   517511,   95901;\n  10: 0, 109601, 1159880, 5334398, 13934096, 22622179, 23385328, 15037350, 5501032, 876809;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(n-3+(3*n-9)*x+(3*n-9)*x^2+(n-3)*x^3)*P(n-2,x) + (n-1+(2*n-1)*x+n*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 3 + (3*n - 9)*x + (3*n - 9)*x^2 + (n - 3)*x^3)*P[n - 2, x] + (n - 1 + (2*n - 1)*x + n*x^2)*P[n - 1, x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = x; for(n = 3, N, my(i = n); v[i] = (-(n - 3 + (3*n - 9)*x + (3*n - 9)*x^2 + (n - 3)*x^3)*v[i - 2] + (n - 1 + (2*n - 1)*x + n*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cd3918419c5686a8","steps":["x \u21a6 x/(x+1)"],"edges":["e-60865b531221d46e"],"text":"A269951 \u2192 x \u21a6 x/(x+1)"},"headline":"A269951 \u2192 x \u21a6 x/(x+1)","chain":null}
