{"id":"n-a492dc7276777329","revision":1,"first_index":0,"fingerprint":"f6-7cabb77b45806852","name":"mobius_x_over_x_plus_one of Triangle, row sums = number of edges in n-dimensional hypercubes.","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":true,"rows":[["1"],["1","4"],["1","8","12"],["1","12","36","32"],["1","16","72","128","80"],["1","20","120","320","400","192"],["1","24","180","640","1200","1152","448"],["1","28","252","1120","2800","4032","3136","1024"],["1","32","336","1792","5600","10752","12544","8192","2304"],["1","36","432","2688","10080","24192","37632","36864","20736","5120"],["1","40","540","3840","16800","48384","94080","122880","103680","51200","11264"],["1","44","660","5280","26400","88704","206976","337920","380160","281600","123904","24576"],["1","48","792","7040","39600","152064","413952","811008","1140480","1126400","743424","294912","53248"],["1","52","936","9152","57200","247104","768768","1757184","2965248","3660800","3221504","1916928","692224","114688"],["1","56","1092","11648","80080","384384","1345344","3514368","6918912","10250240","11275264","8945664","4845568","1605632","245760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","21","81","297","1053","3645","12393","41553","137781","452709","1476225","4782969","15411789","49424013"],"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","-2","-1"],["2","4","2"]]},"terms":[{"coeff":{"coeffs":[["-1","2","11","12","4"],["2","4","-2","-8","-4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","10","18","14","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163541,"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-c033b30f4e233ccb"]},{"id":163542,"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-c033b30f4e233ccb"]},{"id":163543,"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-c033b30f4e233ccb"]}],"region_exact":{},"incoming_edges":[{"id":"e-c033b30f4e233ccb","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-a492dc7276777329","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":"fa288ab8776b3bc8618fdc1d85fd9c7fccf7ee4805beebff4e2ff2ef13514342","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-6202e291a2627bce","role":"sequence","name":"Triangle, row sums = number of edges in n-dimensional hypercubes.","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-616d265f2f37e44a","kind":"candidate_equal","node_a_id":"n-a492dc7276777329","node_b_id":"n-f38c68e75ec69849","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-f38c68e75ec69849","other":{"id":"n-f38c68e75ec69849","name":"theta_plus_one of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-7cabb77b45806852","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4"],["1","8","12"]],"card_rows":[["1"],["1","4"],["1","8","12"],["1","12","36","32"]],"row_sums_preview":["1","5","21","81","297","1053","3645","12393"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["\u03b8+1"],"edges":["e-1108847626c3577f"],"text":"A013609 \u2192 \u03b8+1"},"headline":"A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-7b8c011230aabf19","kind":"fingerprint_related","node_a_id":"n-a492dc7276777329","node_b_id":"n-58281abfb06e824f","fingerprint":"f6-7cabb77b45806852","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7cabb77b45806852"},"other_id":"n-58281abfb06e824f","other":{"id":"n-58281abfb06e824f","name":"theta_plus_one of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-7cabb77b45806852","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["3","12"],["3","24","36"]],"card_rows":[["3"],["3","12"],["3","24","36"],["3","36","108","96"]],"row_sums_preview":["3","15","63","243","891","3159","10935","37179"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["\u03b8+1"],"edges":["e-8d945ebe18ee4c8a"],"text":"A120909 \u2192 \u03b8+1"},"headline":"A120909 \u2192 \u03b8+1","chain":null}},{"id":"l-76a5af2a2b86b2cf","kind":"oeis_match","node_a_id":"n-a492dc7276777329","node_b_id":null,"fingerprint":null,"oeis_id":"A081038","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":"A081038","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A081038","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-a492dc7276777329","oeis_id":"A081038","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":10830}],"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,  12;\n  3: 1, 12,  36,   32;\n  4: 1, 16,  72,  128,    80;\n  5: 1, 20, 120,  320,   400,   192;\n  6: 1, 24, 180,  640,  1200,  1152,   448;\n  7: 1, 28, 252, 1120,  2800,  4032,  3136,  1024;\n  8: 1, 32, 336, 1792,  5600, 10752, 12544,  8192,  2304;\n  9: 1, 36, 432, 2688, 10080, 24192, 37632, 36864, 20736, 5120;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (2*n-1+(4*n+2)*x-(2*n-11)*x^2-(8*n-12)*x^3-(4*n-4)*x^4)*P(n-1,x) + (2*x+10*x^2+18*x^3+14*x^4+4*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[((2*n - 1 + (4*n + 2)*x - (2*n - 11)*x^2 - (8*n - 12)*x^3 - (4*n - 4)*x^4)*P[n - 1, x] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*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] = ((2*n - 1 + (4*n + 2)*x - (2*n - 11)*x^2 - (8*n - 12)*x^3 - (4*n - 4)*x^4)*v[i - 1] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*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-c8c19eba9d68881f","steps":["2\u03b8+1","x \u21a6 x/(x+1)"],"edges":["e-c4ea45619bc7f60a","e-c033b30f4e233ccb"],"text":"A007318 \u2192 2\u03b8+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A007318 \u2192 2\u03b8+1 \u2192 x \u21a6 x/(x+1)","chain":null}