{"id":"n-2d0ce1c0452a8c5a","revision":0,"first_index":0,"fingerprint":"f6-c761722d499074dd","name":"mobius_x_over_x_plus_one of basis_map_A013609 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle 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"],["4","8"],["16","64","60"],["64","384","736","448"],["256","2048","5952","7424","3344"],["1024","10240","39936","75776","69824","24960"],["4096","49152","240640","614400","861696","628736","186304"],["16384","229376","1351680","4341760","8202240","9105408","5495808","1390592"],["65536","1048576","7225344","27983872","66580480","99581952","91379712","47013888","10379520"],["262144","4718592","37224448","168820736","484786176","913604608","1129365504","882671616","395645952","77473792"],["1048576","20971520","186384384","968884224","3260809216","7420772352","11559976960","12167282688","8278437888","3287007232","578272256"],["4194304","92274688","912261120","5347737600","20645412864","55094280192","103666941952","137494528000","125934551040","75844845568","27026563072","4316282880"],["16777216","402653184","4383047680","28605153280","124618014720","381664886784","842378772480","1349630558208","1557454520320","1262186659840","681751265280","220329607168","32217174016"],["67108864","1744830464","20736638976","149115895808","723664240640","2502212714496","6340252532736","11917175291904","16612194779136","16958404689920","12320760135680","6031949168640","1783399301120","240472260608"],["268435456","7516192768","96838090752","760746082304","4070051938304","15683744169984","44880239788032","96864061882368","158416666361856","195332654759936","178720459980800","117645131120640","52662185492480","14347765022720","1794909388800"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","140","1632","19024","221760","2585024","30133248","351258880","4094573568","47729847296","556379873280","6485639090176","75602149588992","881283238707200"],"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":[["0","0","0"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["0","-4","-14","-16","-6"],["4","20","34","24","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-12","-24","-20","-6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183168,"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-2be5f4d4473cb60f"]},{"id":183169,"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-2be5f4d4473cb60f"]},{"id":183170,"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-2be5f4d4473cb60f"]}],"region_exact":{},"incoming_edges":[{"id":"e-2be5f4d4473cb60f","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-2d0ce1c0452a8c5a","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":"e64fc7e418d58e79024a8a5d26c3f06ef180829661fc14ff91055f1575e5635b","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-3d0ce3ba50a1aaca","role":"sequence","name":"basis_map_A013609 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle 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":[{"id":"l-7a4328c5cc6d93b6","kind":"fingerprint_related","node_a_id":"n-5c55f31aa1da8314","node_b_id":"n-2d0ce1c0452a8c5a","fingerprint":"f6-c761722d499074dd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c761722d499074dd"},"other_id":"n-5c55f31aa1da8314","other":{"id":"n-5c55f31aa1da8314","name":"shift_x_plus_one of shift_x_plus_one of Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).","first_index":1,"fingerprint":"f6-c761722d499074dd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["15","16","4"]],"card_rows":[["1"],["4","2"],["15","16","4"],["56","92","48","8"]],"row_sums_preview":["1","6","35","204","1189","6930","40391","235416"],"regions":["(-3, -1)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a2d7d802498a9d22","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-5342624c2722e8d8","e-3b84462f43b7ccde"],"text":"A053117 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A053117 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-b80954391f9e12b3","kind":"fingerprint_related","node_a_id":"n-3373029d649348a1","node_b_id":"n-2d0ce1c0452a8c5a","fingerprint":"f6-c761722d499074dd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c761722d499074dd"},"other_id":"n-3373029d649348a1","other":{"id":"n-3373029d649348a1","name":"reversal of basis_map_A013609 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","first_index":0,"fingerprint":"f6-c761722d499074dd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["4","16","15"]],"card_rows":[["1"],["2","4"],["4","16","15"],["8","48","92","56"]],"row_sums_preview":["1","6","35","204","1189","6930","40391","235416"],"regions":["(-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f287bed89e07609f","steps":["basis A013609","reverse"],"edges":["e-7797a63e71923f20","e-2d599e44c7096066"],"text":"A125662 \u2192 basis A013609 \u2192 reverse"},"headline":"A125662 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-e80ab30a794b7a3f","kind":"fingerprint_related","node_a_id":"n-7d7dad3bd149bf81","node_b_id":"n-2d0ce1c0452a8c5a","fingerprint":"f6-c761722d499074dd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c761722d499074dd"},"other_id":"n-7d7dad3bd149bf81","other":{"id":"n-7d7dad3bd149bf81","name":"basis_map_A013609 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","first_index":0,"fingerprint":"f6-c761722d499074dd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["15","16","4"]],"card_rows":[["1"],["4","2"],["15","16","4"],["56","92","48","8"]],"row_sums_preview":["1","6","35","204","1189","6930","40391","235416"],"regions":["(-inf, 0]","(-3, -1)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f287bed89e07609f","steps":["basis A013609"],"edges":["e-7797a63e71923f20"],"text":"A125662 \u2192 basis A013609"},"headline":"A125662 \u2192 basis A013609","chain":null}}],"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:      4,       8;\n  2:     16,      64,       60;\n  3:     64,     384,      736,       448;\n  4:    256,    2048,     5952,      7424,      3344;\n  5:   1024,   10240,    39936,     75776,     69824,     24960;\n  6:   4096,   49152,   240640,    614400,    861696,    628736,     186304;\n  7:  16384,  229376,  1351680,   4341760,   8202240,   9105408,    5495808,   1390592;\n  8:  65536, 1048576,  7225344,  27983872,  66580480,  99581952,   91379712,  47013888,  10379520;\n  9: 262144, 4718592, 37224448, 168820736, 484786176, 913604608, 1129365504, 882671616, 395645952, 77473792;\n","recurrence_sentence":"The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (4*n+(20*n-4)*x+(34*n-14)*x^2+(24*n-16)*x^3+(6*n-6)*x^4)*P(n-1,x) - (2*x+12*x^2+24*x^3+20*x^4+6*x^5)*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[((4*n + (20*n - 4)*x + (34*n - 14)*x^2 + (24*n - 16)*x^3 + (6*n - 6)*x^4)*P[n - 1, x] - (2*x + 12*x^2 + 24*x^3 + 20*x^4 + 6*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*x^2)]];\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] = ((4*n + (20*n - 4)*x + (34*n - 14)*x^2 + (24*n - 16)*x^3 + (6*n - 6)*x^4)*v[i - 1] - (2*x + 12*x^2 + 24*x^3 + 20*x^4 + 6*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*x^2)); 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-2245b35ec7cd4857","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-11b86d48c22b376e","e-2be5f4d4473cb60f"],"text":"A091044 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A091044 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}