{"id":"n-968c9f27d7b2c4c9","revision":1,"first_index":0,"fingerprint":"f6-c8aa5aaa0c2f2cd0","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","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"],["1","3"],["1","8","11"],["1","15","51","45"],["1","24","146","304","197"],["1","35","330","1190","1765","903"],["1","48","645","3520","8835","10128","4279"],["1","63","1141","8715","32515","62013","57799","20793"],["1","80","1876","19040","98014","274064","419524","329024","103049"],["1","99","2916","37884","256158","977130","2169300","2766060","1871145","518859"],["1","120","4335","70080","600810","2982000","8949150","16406400","17897925","10639320","2646723"],["1","143","6215","122265","1293930","8078070","31298190","76984050","119893125","114171915","60511803","13648869"],["1","168","8646","203280","2600235","19902960","96347460","303969600","631196775","853038120","720268758","344343024","71039373"],["1","195","11726","324610","4933643","45376617","267752628","1049448972","2779811463","4982835429","5941240734","4503830994","1960751949","372693519"],["1","224","15561","500864","8917909","96960864","683984301","3251449344","10647607971","24232240032","38150787675","40666254720","27960063495","11172756000","1968801519"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","20","112","672","4224","27456","183040","1244672","8599552","60196864","426008576","3042918400","21909012480","158840340480"],"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":[["2","4","2"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["2","8","8","0","-2"],["1","7","13","9","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","4","0","-4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165284,"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-c838eb74321cec2f"]},{"id":165285,"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-c838eb74321cec2f"]},{"id":165286,"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-c838eb74321cec2f"]}],"region_exact":{},"incoming_edges":[{"id":"e-c838eb74321cec2f","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-968c9f27d7b2c4c9","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":"8c7adaa31e35eb5097e3344bd8df7f14431c551d08fa330481cc0d4818c93925","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-8ce61d302cec295b","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","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-1017d30842b71558","kind":"fingerprint_related","node_a_id":"n-968c9f27d7b2c4c9","node_b_id":"n-25065a706ad43d32","fingerprint":"f6-c8aa5aaa0c2f2cd0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c8aa5aaa0c2f2cd0"},"other_id":"n-25065a706ad43d32","other":{"id":"n-25065a706ad43d32","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.","first_index":1,"fingerprint":"f6-c8aa5aaa0c2f2cd0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["6","2"],["22","16","2"]],"card_rows":[["2"],["6","2"],["22","16","2"],["90","102","30","2"]],"row_sums_preview":["2","8","40","224","1344","8448","54912","366080"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a30765140629bd5","steps":["x \u21a6 x+1"],"edges":["e-5ebd2cd7e542e718"],"text":"A114655 \u2192 x \u21a6 x+1"},"headline":"A114655 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-168c1da812505968","kind":"fingerprint_related","node_a_id":"n-6ad396ce174787e3","node_b_id":"n-968c9f27d7b2c4c9","fingerprint":"f6-c8aa5aaa0c2f2cd0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c8aa5aaa0c2f2cd0"},"other_id":"n-6ad396ce174787e3","other":{"id":"n-6ad396ce174787e3","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","first_index":1,"fingerprint":"f6-c8aa5aaa0c2f2cd0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["11","8","1"]],"card_rows":[["1"],["3","1"],["11","8","1"],["45","51","15","1"]],"row_sums_preview":["1","4","20","112","672","4224","27456","183040"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0d75f6272d2a2a84","steps":["basis A007318"],"edges":["e-38a7985c4c1aa2e3"],"text":"A114656 \u2192 basis A007318"},"headline":"A114656 \u2192 basis A007318","chain":null}},{"id":"l-134c1b8cd816d87d","kind":"oeis_match","node_a_id":"n-968c9f27d7b2c4c9","node_b_id":null,"fingerprint":null,"oeis_id":"A003645","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":"A003645","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A003645","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-968c9f27d7b2c4c9","oeis_id":"A003645","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":18469}],"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,  3;\n  2: 1,  8,   11;\n  3: 1, 15,   51,    45;\n  4: 1, 24,  146,   304,    197;\n  5: 1, 35,  330,  1190,   1765,    903;\n  6: 1, 48,  645,  3520,   8835,  10128,    4279;\n  7: 1, 63, 1141,  8715,  32515,  62013,   57799,   20793;\n  8: 1, 80, 1876, 19040,  98014, 274064,  419524,  329024,  103049;\n  9: 1, 99, 2916, 37884, 256158, 977130, 2169300, 2766060, 1871145, 518859;\n","recurrence_sentence":"The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (n+2+(7*n+8)*x+(13*n+8)*x^2+9*n*x^3+(2*n-2)*x^4)*P(n-1,x) + (2*x+4*x^2-4*x^4-2*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[((n + 2 + (7*n + 8)*x + (13*n + 8)*x^2 + 9*n*x^3 + (2*n - 2)*x^4)*P[n - 1, x] + (2*x + 4*x^2 - 4*x^4 - 2*x^5)*D[P[n - 1, x], x])/(n + 2 + (2*n + 4)*x + (n + 2)*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] = ((n + 2 + (7*n + 8)*x + (13*n + 8)*x^2 + 9*n*x^3 + (2*n - 2)*x^4)*v[i - 1] + (2*x + 4*x^2 - 4*x^4 - 2*x^5)*deriv(v[i - 1], x))/(n + 2 + (2*n + 4)*x + (n + 2)*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-8ce61d302cec295b","steps":["x \u21a6 x/(x+1)"],"edges":["e-c838eb74321cec2f"],"text":"A114687 \u2192 x \u21a6 x/(x+1)"},"headline":"A114687 \u2192 x \u21a6 x/(x+1)","chain":null}