{"id":"n-9af5c0b0557bf778","revision":1,"first_index":0,"fingerprint":"f6-3b2315a230a09078","name":"mobius_x_over_x_plus_one of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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","4"],["1","8","10"],["1","12","36","28"],["1","16","78","136","76"],["1","20","136","388","476","208"],["1","24","210","848","1680","1584","568"],["1","28","300","1580","4424","6672","5104","1552"],["1","32","406","2648","9700","20576","24976","16064","4240"],["1","36","528","4116","18756","51984","88320","89568","49680","11584"],["1","40","666","6048","33096","114192","252840","357312","310896","151552","31648"],["1","44","820","8508","54480","226128","622704","1143600","1381296","1051840","457280","86464"],["1","48","990","11560","84924","413376","1369536","3123072","4888800","5150848","3485856","1367424","236224"],["1","52","1176","15268","126700","709200","2757504","7559808","14667552","19978240","18653824","11357376","4058560","645376"],["1","56","1378","19696","182336","1155568","5172504","16640640","38772384","65337088","78689600","65938432","36479872","11969792","1763200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","19","77","307","1229","4915","19661","78643","314573","1258291","5033165","20132659","80530637","322122547"],"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","4"]],"recurrence":{"denominator":{"coeffs":[["-1","-3","-3","-1"],["1","3","3","1"]]},"terms":[{"coeff":{"coeffs":[["-1","13","87","197","214","114","24"],["-3","-41","-171","-333","-338","-174","-36"],["4","28","84","136","124","60","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","19","142","426","664","571","258","48"],["-1","-18","-98","-256","-369","-302","-132","-24"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","17","82","190","245","181","72","12"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163892,"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-b8a3e660dc9132dc"]},{"id":163893,"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-b8a3e660dc9132dc"]},{"id":163894,"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-b8a3e660dc9132dc"]}],"region_exact":{},"incoming_edges":[{"id":"e-b8a3e660dc9132dc","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-9af5c0b0557bf778","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":"7c917b01eeeab01719f7b749b657ba4c972416cd20569fe89e69c9d26bd44377","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-eac80c2bf8e566f5","role":"sequence","name":"Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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-659bf241c5daf38b","kind":"oeis_match","node_a_id":"n-9af5c0b0557bf778","node_b_id":null,"fingerprint":null,"oeis_id":"A108981","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A108981","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A108981","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-9af5c0b0557bf778","oeis_id":"A108981","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":26,"oeis_terms_available":26,"term_count":200,"raw_size":12987}],"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,  4;\n  2: 1,  8,  10;\n  3: 1, 12,  36,   28;\n  4: 1, 16,  78,  136,    76;\n  5: 1, 20, 136,  388,   476,   208;\n  6: 1, 24, 210,  848,  1680,  1584,   568;\n  7: 1, 28, 300, 1580,  4424,  6672,  5104,  1552;\n  8: 1, 32, 406, 2648,  9700, 20576, 24976, 16064,  4240;\n  9: 1, 36, 528, 4116, 18756, 51984, 88320, 89568, 49680, 11584;\n","recurrence_sentence":"The row polynomials satisfy (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (4*n^2-3*n-1+(28*n^2-41*n+13)*x+(84*n^2-171*n+87)*x^2+(136*n^2-333*n+197)*x^3+(124*n^2-338*n+214)*x^4+(60*n^2-174*n+114)*x^5+(12*n^2-36*n+24)*x^6)*P(n-1,x) - (n+(18*n-19)*x+(98*n-142)*x^2+(256*n-426)*x^3+(369*n-664)*x^4+(302*n-571)*x^5+(132*n-258)*x^6+(24*n-48)*x^7)*d/dx P(n-1,x) + (x+17*x^2+82*x^3+190*x^4+245*x^5+181*x^6+72*x^7+12*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 3*n - 1 + (28*n^2 - 41*n + 13)*x + (84*n^2 - 171*n + 87)*x^2 + (136*n^2 - 333*n + 197)*x^3 + (124*n^2 - 338*n + 214)*x^4 + (60*n^2 - 174*n + 114)*x^5 + (12*n^2 - 36*n + 24)*x^6)*P[n - 1, x] - (n + (18*n - 19)*x + (98*n - 142)*x^2 + (256*n - 426)*x^3 + (369*n - 664)*x^4 + (302*n - 571)*x^5 + (132*n - 258)*x^6 + (24*n - 48)*x^7)*D[P[n - 1, x], x] + (x + 17*x^2 + 82*x^3 + 190*x^4 + 245*x^5 + 181*x^6 + 72*x^7 + 12*x^8)*D[P[n - 1, x], {x, 2}])/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 3*n - 1 + (28*n^2 - 41*n + 13)*x + (84*n^2 - 171*n + 87)*x^2 + (136*n^2 - 333*n + 197)*x^3 + (124*n^2 - 338*n + 214)*x^4 + (60*n^2 - 174*n + 114)*x^5 + (12*n^2 - 36*n + 24)*x^6)*v[i - 1] - (n + (18*n - 19)*x + (98*n - 142)*x^2 + (256*n - 426)*x^3 + (369*n - 664)*x^4 + (302*n - 571)*x^5 + (132*n - 258)*x^6 + (24*n - 48)*x^7)*deriv(v[i - 1], x) + (x + 17*x^2 + 82*x^3 + 190*x^4 + 245*x^5 + 181*x^6 + 72*x^7 + 12*x^8)*deriv(deriv(v[i - 1], x), x))/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)); 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-eac80c2bf8e566f5","steps":["x \u21a6 x/(x+1)"],"edges":["e-b8a3e660dc9132dc"],"text":"A249250 \u2192 x \u21a6 x/(x+1)"},"headline":"A249250 \u2192 x \u21a6 x/(x+1)","chain":null}
