{"id":"n-2a5188b3bacc3f29","revision":0,"first_index":0,"fingerprint":"f6-c8aa5aaa0c2f2cd0","name":"shift_x_plus_one of reversal 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"],["3","1"],["11","8","1"],["45","51","15","1"],["197","304","146","24","1"],["903","1765","1190","330","35","1"],["4279","10128","8835","3520","645","48","1"],["20793","57799","62013","32515","8715","1141","63","1"],["103049","329024","419524","274064","98014","19040","1876","80","1"],["518859","1871145","2766060","2169300","977130","256158","37884","2916","99","1"],["2646723","10639320","17897925","16406400","8949150","2982000","600810","70080","4335","120","1"],["13648869","60511803","114171915","119893125","76984050","31298190","8078070","1293930","122265","6215","143","1"],["71039373","344343024","720268758","853038120","631196775","303969600","96347460","19902960","2600235","203280","8646","168","1"],["372693519","1960751949","4503830994","5941240734","4982835429","2779811463","1049448972","267752628","45376617","4933643","324610","11726","195","1"],["1968801519","11172756000","27960063495","40666254720","38150787675","24232240032","10647607971","3251449344","683984301","96960864","8917909","500864","15561","224","1"]],"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:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","0"],["5","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176571,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-4d97c9c0745dee04"]},{"id":176572,"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-4d97c9c0745dee04"]},{"id":176573,"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-4d97c9c0745dee04"]},{"id":176574,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-4d97c9c0745dee04"]}],"region_exact":{},"incoming_edges":[{"id":"e-4d97c9c0745dee04","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-2a5188b3bacc3f29","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"d3ddacfca96e83d1b41a6fa0601eebccc75f504036bf52fe78880fc2d94303d5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1b0fc3e5a40ede1a","role":"sequence","name":"reversal 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).","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-175c0b6bb7c23a3f","kind":"fingerprint_related","node_a_id":"n-2a5188b3bacc3f29","node_b_id":"n-57c77e76c5c7eb67","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-57c77e76c5c7eb67","other":{"id":"n-57c77e76c5c7eb67","name":"reversal of 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"],["1","3"],["1","8","11"]],"card_rows":[["1"],["1","3"],["1","8","11"],["1","15","51","45"]],"row_sums_preview":["1","4","20","112","672","4224","27456","183040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0d75f6272d2a2a84","steps":["basis A007318","reverse"],"edges":["e-38a7985c4c1aa2e3","e-7ea83929b8df6dc6"],"text":"A114656 \u2192 basis A007318 \u2192 reverse"},"headline":"A114656 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-23e5d036e5d98a83","kind":"fingerprint_related","node_a_id":"n-2a5188b3bacc3f29","node_b_id":"n-7857580ebfc5510f","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-7857580ebfc5510f","other":{"id":"n-7857580ebfc5510f","name":"mobius_x_over_x_plus_one of reversal 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"],["2","6"],["2","16","22"]],"card_rows":[["2"],["2","6"],["2","16","22"],["2","30","102","90"]],"row_sums_preview":["2","8","40","224","1344","8448","54912","366080"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a30765140629bd5","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-a35c7072fcaf677b","e-a3117fe3a3841740"],"text":"A114655 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A114655 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-54022897b563abbf","kind":"fingerprint_related","node_a_id":"n-2a5188b3bacc3f29","node_b_id":"n-31d20c36c762bb76","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-31d20c36c762bb76","other":{"id":"n-31d20c36c762bb76","name":"mobius_x_over_x_plus_one of reversal 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"],["1","3"],["1","8","11"]],"card_rows":[["1"],["1","3"],["1","8","11"],["1","15","51","45"]],"row_sums_preview":["1","4","20","112","672","4224","27456","183040"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0d75f6272d2a2a84","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-87267df0bda73e27","e-b524faa27b27477b"],"text":"A114656 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A114656 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-598b02068dcc6097","kind":"fingerprint_related","node_a_id":"n-25065a706ad43d32","node_b_id":"n-2a5188b3bacc3f29","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-59d73053c3a4c02d","kind":"fingerprint_related","node_a_id":"n-6ad396ce174787e3","node_b_id":"n-2a5188b3bacc3f29","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-9027e96bd8e7c3d7","kind":"fingerprint_related","node_a_id":"n-2a5188b3bacc3f29","node_b_id":"n-76d02a8ed88dcf78","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-76d02a8ed88dcf78","other":{"id":"n-76d02a8ed88dcf78","name":"reversal of 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"],["2","6"],["2","16","22"]],"card_rows":[["2"],["2","6"],["2","16","22"],["2","30","102","90"]],"row_sums_preview":["2","8","40","224","1344","8448","54912","366080"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a30765140629bd5","steps":["x \u21a6 x+1","reverse"],"edges":["e-5ebd2cd7e542e718","e-3883f0f1d619c95a"],"text":"A114655 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A114655 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-e763deef2322e1fa","kind":"fingerprint_related","node_a_id":"n-968c9f27d7b2c4c9","node_b_id":"n-2a5188b3bacc3f29","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-968c9f27d7b2c4c9","other":{"id":"n-968c9f27d7b2c4c9","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).","first_index":0,"fingerprint":"f6-c8aa5aaa0c2f2cd0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","8","11"]],"card_rows":[["1"],["1","3"],["1","8","11"],["1","15","51","45"]],"row_sums_preview":["1","4","20","112","672","4224","27456","183040"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"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}}],"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:      3,       1;\n  2:     11,       8,       1;\n  3:     45,      51,      15,       1;\n  4:    197,     304,     146,      24,      1;\n  5:    903,    1765,    1190,     330,     35,      1;\n  6:   4279,   10128,    8835,    3520,    645,     48,     1;\n  7:  20793,   57799,   62013,   32515,   8715,   1141,    63,    1;\n  8: 103049,  329024,  419524,  274064,  98014,  19040,  1876,   80,  1;\n  9: 518859, 1871145, 2766060, 2169300, 977130, 256158, 37884, 2916, 99, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (5*n+4+3*n*x)*P(n-1,x) + (2-2*x^2)*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[((5*n + 4 + 3*n*x)*P[n - 1, x] + (2 - 2*x^2)*D[P[n - 1, x], x])/(n + 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] = ((5*n + 4 + 3*n*x)*v[i - 1] + (2 - 2*x^2)*deriv(v[i - 1], x))/(n + 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":["reverse","x \u21a6 x+1"],"edges":["e-0ad5e2ffb4ffd4ba","e-4d97c9c0745dee04"],"text":"A114687 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A114687 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}