{"id":"n-46fcdb90dd7e322d","revision":1,"first_index":0,"fingerprint":"f6-e49234ad2c21a445","name":"mobius_x_over_x_plus_one of Triangle read by rows: A008277 * A007318.","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","3"],["5","15","11"],["15","67","98","47"],["52","307","670","641","227"],["203","1486","4299","6146","4344","1215"],["877","7648","27501","52204","55191","30823","7107"],["4140","41847","179612","424411","596399","498541","229606","44959"],["21147","242857","1210098","3417413","5983741","6653261","4588620","1795165","305091"],["115975","1490395","8448371","27728366","58078061","80519503","73904431","43312372","14710196","2206399"],["678570","9642157","61228414","228830560","557467576","925127489","1059300824","826511942","420621997","126099219","16913987"],["4213597","65566810","460792826","1930750680","5359756709","10351433940","14194530888","13821842398","9367509843","4208932311","1128545538","136823263"],["27644437","467343692","3599589830","16704086156","52019935595","114543380698","182877096480","213337877026","180497020298","108026987023","43418893178","10523995721","1163490499"],["190899322","3482851087","29163988944","148415787378","512171169652","1265763672317","2304978474798","3131819329640","3175327926055","2372990966204","1270678430163","461737002390","102064972112","10366252031"],["1382958545","27075644687","244823629233","1355248327354","5131334918139","14058689437649","28744974030631","44561462406460","52637287166179","47151038205270","31532420936303","15267671036864","5060097708979","1027627654399","96491364675"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","31","227","1897","17693","181351","2019515","24217393","310514069","4232422735","61030698803","927207340633","14789151722093","246897625425367"],"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"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["3","9","8","2"],["-1","-4","-5","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","5","9","7","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165562,"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-426e25db4b42e73d"]},{"id":165563,"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-426e25db4b42e73d"]},{"id":165564,"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-426e25db4b42e73d"]}],"region_exact":{},"incoming_edges":[{"id":"e-426e25db4b42e73d","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-46fcdb90dd7e322d","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":"1cf3bfe852b8f001b541a5fbb3cbf4605c362ef44c15e70fddc220232614c2af","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-bdae5aa3d1cac0ec","role":"sequence","name":"Triangle read by rows: A008277 * A007318.","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:      2,       3;\n  2:      5,      15,      11;\n  3:     15,      67,      98,       47;\n  4:     52,     307,     670,      641,      227;\n  5:    203,    1486,    4299,     6146,     4344,     1215;\n  6:    877,    7648,   27501,    52204,    55191,    30823,     7107;\n  7:   4140,   41847,  179612,   424411,   596399,   498541,   229606,    44959;\n  8:  21147,  242857, 1210098,  3417413,  5983741,  6653261,  4588620,  1795165,   305091;\n  9: 115975, 1490395, 8448371, 27728366, 58078061, 80519503, 73904431, 43312372, 14710196, 2206399;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(n-3+(4*n-9)*x+(5*n-8)*x^2+(2*n-2)*x^3)*P(n-1,x) + (1+5*x+9*x^2+7*x^3+2*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 3 + (4*n - 9)*x + (5*n - 8)*x^2 + (2*n - 2)*x^3)*P[n - 1, x] + (1 + 5*x + 9*x^2 + 7*x^3 + 2*x^4)*D[P[n - 1, x], x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(n - 3 + (4*n - 9)*x + (5*n - 8)*x^2 + (2*n - 2)*x^3)*v[i - 1] + (1 + 5*x + 9*x^2 + 7*x^3 + 2*x^4)*deriv(v[i - 1], x))/(1 + x)); 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-c8c19eba9d68881f","steps":["basis A049020","x \u21a6 x/(x+1)"],"edges":["e-3c112fda27f20cb0","e-426e25db4b42e73d"],"text":"A007318 \u2192 basis A049020 \u2192 x \u21a6 x/(x+1)"},"headline":"A007318 \u2192 basis A049020 \u2192 x \u21a6 x/(x+1)","chain":null}