{"id":"n-f45c2041d5d8c4e2","revision":0,"first_index":1,"fingerprint":"f6-44f2cdf251913dc5","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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","3"],["3","14","15"],["7","47","101","69"],["17","148","470","644","321"],["41","439","1842","3782","3797","1491"],["99","1258","6553","17900","27029","21386","6927"],["239","3515","21859","74479","150109","178913","116753","32181"],["577","9640","69660","284280","716422","1141464","1122716","623240","149505"],["1393","26059","214516","1019660","3083566","6151402","8094020","6773276","3270953","694563"],["3363","69638","643219","3489224","12308294","29497060","48631486","54461128","39645623","16940678","3226767"],["8119","184359","1887921","11507225","46379238","129771942","257201538","361051762","351781875","226562051","86807357","14990757"],["19601","484284","5445138","36837148","166982223","534258744","1237035452","2088387928","2551155391","2199158156","1269789266","440940044","69643329"],["47321","1263807","15477126","115066834","579384939","2086256949","5526532900","10904091404","16023456031","17319782713","13384612342","7002943490","2223431933","323545587"],["114243","3279586","43450557","352112724","1949703443","7802752734","23273110693","52553620376","90279091137","117402333054","113768243599","79661907924","38101409129","11142212354","1503112335"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","32","224","1600","11392","81152","578048","4117504","29329408","208916480","1488134144","10600136704","75505893376","537836453888"],"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"],["1","3"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["1","5","7","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","6","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177050,"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-58ebb55fac0d1901"]},{"id":177051,"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-58ebb55fac0d1901"]},{"id":177052,"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-58ebb55fac0d1901"]}],"region_exact":{},"incoming_edges":[{"id":"e-58ebb55fac0d1901","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-f45c2041d5d8c4e2","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":"1d32b792fb57a0515fabe9b1e854dd79ecbe6b907d3770059e558edbf55fad5b","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-84dfb3e427b94d98","role":"sequence","name":"basis_map_A013609 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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":[],"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,     3;\n   3:    3,    14,     15;\n   4:    7,    47,    101,      69;\n   5:   17,   148,    470,     644,     321;\n   6:   41,   439,   1842,    3782,    3797,    1491;\n   7:   99,  1258,   6553,   17900,   27029,   21386,    6927;\n   8:  239,  3515,  21859,   74479,  150109,  178913,  116753,   32181;\n   9:  577,  9640,  69660,  284280,  716422, 1141464, 1122716,  623240,  149505;\n  10: 1393, 26059, 214516, 1019660, 3083566, 6151402, 8094020, 6773276, 3270953, 694563;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (1+5*x+7*x^2+3*x^3)*P(n-2,x) + (2+6*x+4*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[((1 + 5*x + 7*x^2 + 3*x^3)*P[n - 2, x] + (2 + 6*x + 4*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] = 1 + 3*x; for(n = 3, N, my(i = n); v[i] = ((1 + 5*x + 7*x^2 + 3*x^3)*v[i - 2] + (2 + 6*x + 4*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-b0fc620949fb5990","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-778f192f6d6aff27","e-58ebb55fac0d1901"],"text":"A122542 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A122542 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}
