{"id":"n-239738a029785d3a","revision":1,"first_index":1,"fingerprint":"f6-c43ba15ffe3d8a71","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","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"],["1","5"],["1","12","21"],["1","21","84","84"],["1","32","216","480","330"],["1","45","450","1650","2475","1287"],["1","60","825","4400","10725","12012","5005"],["1","77","1386","10010","35035","63063","56056","19448"],["1","96","2184","20384","95550","244608","346528","254592","75582"],["1","117","3276","38220","229320","779688","1559376","1813968","1133730","293930"],["1","140","4725","67200","499800","2159136","5697720","9302400","9157050","4974200","1144066"],["1","165","6600","112200","1009800","5372136","17907120","38372400","52762050","44945450","21573816","4457400"],["1","192","8976","179520","1918620","12279168","50139936","135070848","242705430","287650880","215738160","92713920","17383860"],["1","221","11934","277134","3464175","26189163","128035908","420689412","946551177","1460727125","1519156210","1016955810","395482815","67863915"],["1","252","15561","414960","5990985","52720668","303143841","1187828928","3247969725","6255349100","8444721285","7816601520","4722530085","1676637900","265182525"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","34","190","1059","5908","33028","185076","1039525","5851626","33006438","186519138","1055789511","5985405000","33979107336"],"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"]],"recurrence":{"denominator":{"coeffs":[["-2","-4","-2"],["1","2","1"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["-2","-2","2","2"],["1","7","11","5"],["1","4","5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","3","3","1"],["0","2","6","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164800,"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-20496aa59e8ba6c2"]},{"id":164801,"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-20496aa59e8ba6c2"]},{"id":164802,"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-20496aa59e8ba6c2"]}],"region_exact":{},"incoming_edges":[{"id":"e-20496aa59e8ba6c2","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-239738a029785d3a","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":"5305359bbb3436c70a4c5778c327df7e8f467d5c0a3295c20a5d33ed90d7a0a7","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-a2304b34e998691c","role":"sequence","name":"Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","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-8b34eb625016e9f2","kind":"oeis_match","node_a_id":"n-239738a029785d3a","node_b_id":null,"fingerprint":null,"oeis_id":"A049608","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A049608","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A049608","relation":"match_row_sums"}],"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: 1,   5;\n   3: 1,  12,   21;\n   4: 1,  21,   84,    84;\n   5: 1,  32,  216,   480,    330;\n   6: 1,  45,  450,  1650,   2475,   1287;\n   7: 1,  60,  825,  4400,  10725,  12012,    5005;\n   8: 1,  77, 1386, 10010,  35035,  63063,   56056,   19448;\n   9: 1,  96, 2184, 20384,  95550, 244608,  346528,  254592,   75582;\n  10: 1, 117, 3276, 38220, 229320, 779688, 1559376, 1813968, 1133730, 293930;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n-2+(2*n^2+2*n-4)*x+(n^2+n-2)*x^2)*P(n,x) = (n^2+n-2+(4*n^2+7*n-2)*x+(5*n^2+11*n+2)*x^2+(2*n^2+5*n+2)*x^3)*P(n-1,x) + ((2*n+1)*x+(6*n+3)*x^2+(6*n+3)*x^3+(2*n+1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + n - 2 + (4*n^2 + 7*n - 2)*x + (5*n^2 + 11*n + 2)*x^2 + (2*n^2 + 5*n + 2)*x^3)*P[n - 1, x] + ((2*n + 1)*x + (6*n + 3)*x^2 + (6*n + 3)*x^3 + (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^2 + n - 2 + (2*n^2 + 2*n - 4)*x + (n^2 + n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^2 + n - 2 + (4*n^2 + 7*n - 2)*x + (5*n^2 + 11*n + 2)*x^2 + (2*n^2 + 5*n + 2)*x^3)*v[i - 1] + ((2*n + 1)*x + (6*n + 3)*x^2 + (6*n + 3)*x^3 + (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^2 + n - 2 + (2*n^2 + 2*n - 4)*x + (n^2 + n - 2)*x^2)); 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-a2304b34e998691c","steps":["x \u21a6 x/(x+1)"],"edges":["e-20496aa59e8ba6c2"],"text":"A062145 \u2192 x \u21a6 x/(x+1)"},"headline":"A062145 \u2192 x \u21a6 x/(x+1)","chain":null}
