{"id":"n-27eab39bb9297ea0","revision":0,"first_index":1,"fingerprint":"f6-36a7872f2b6a1747","name":"mobius_x_over_x_plus_one of Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, 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"],["1","2"],["1","6","8"],["1","14","50","52"],["1","30","214","552","472"],["1","62","784","3740","7576","5504"],["1","126","2652","20736","74316","124128","78416"],["1","254","8574","102900","578460","1660776","2361824","1320064"],["1","510","26978","477144","3938460","17239896","41295200","51153888","25637824"],["1","1022","83468","2117260","24617148","153806376","550636952","1131695728","1242099040","564275648"],["1","2046","255496","9120432","145207048","1242071208","6222995576","18835853856","33904071392","33410548608","13879795712"],["1","4094","776698","38488868","822272056","9357639872","62933694992","262726030288","688577316608","1102596219136","985967376896","377332365568"],["1","8190","2350542","160087176","4520757264","67041826128","588103933984","3248190463680","11610557596176","26827164180864","38689467601152","31668559570944","11234698041088"],["1","16382","7092552","658972860","24317734416","462659989296","5184106465840","36809016823328","172146151822416","537664234672416","1110687444088320","1457122693003008","1099692789112576","363581406419456"],["1","32766","21359540","2692238688","128681337396","3103228760400","43716245859424","390751704074688","2323345262816528","9410734092227040","26088363172247424","48726156649483008","58631175849038848","41049974904138240","12707452084972544"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","15","117","1269","17667","300375","6032853","139769901","3669332643","107649921375","3490352185077","123938466417189","4783375486212867","199374904588586535"],"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":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["1","4","9","10","4"],["0","0","-2","-4","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","5","9","7","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184281,"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":false,"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-1e4f0d4e4579b71e"]},{"id":184282,"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-1e4f0d4e4579b71e"]},{"id":184283,"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-1e4f0d4e4579b71e"]}],"region_exact":{},"incoming_edges":[{"id":"e-1e4f0d4e4579b71e","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-27eab39bb9297ea0","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":"c63117fc4c8fba3b35b7fe80b71b239c4a0d85a80c4b183af87491cc1a2285ec","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":false,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-5b857304d1bafe7a","role":"sequence","name":"Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0)"}]}],"used_by":[],"used_by_total":0,"links":[],"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: 1,    2;\n   3: 1,    6,     8;\n   4: 1,   14,    50,      52;\n   5: 1,   30,   214,     552,      472;\n   6: 1,   62,   784,    3740,     7576,      5504;\n   7: 1,  126,  2652,   20736,    74316,    124128,     78416;\n   8: 1,  254,  8574,  102900,   578460,   1660776,   2361824,    1320064;\n   9: 1,  510, 26978,  477144,  3938460,  17239896,  41295200,   51153888,   25637824;\n  10: 1, 1022, 83468, 2117260, 24617148, 153806376, 550636952, 1131695728, 1242099040, 564275648;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+4*x-(2*n-9)*x^2-(4*n-10)*x^3-(2*n-4)*x^4)*P(n-1,x) + (x+5*x^2+9*x^3+7*x^4+2*x^5)*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[((1 + 4*x - (2*n - 9)*x^2 - (4*n - 10)*x^3 - (2*n - 4)*x^4)*P[n - 1, x] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*D[P[n - 1, x], x])/(1 + 2*x + 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] = ((1 + 4*x - (2*n - 9)*x^2 - (4*n - 10)*x^3 - (2*n - 4)*x^4)*v[i - 1] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*deriv(v[i - 1], x))/(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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5b857304d1bafe7a","steps":["x \u21a6 x/(x+1)"],"edges":["e-1e4f0d4e4579b71e"],"text":"A112493 \u2192 x \u21a6 x/(x+1)"},"headline":"A112493 \u2192 x \u21a6 x/(x+1)","chain":null}
