{"id":"n-8cac8e8480894b25","revision":0,"first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","name":"shift_x_plus_one of reversal of Triangle of Dyck paths counted by number of long interior inclines.","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":[["2"],["5","3"],["14","16","4"],["42","70","35","5"],["132","288","216","64","6"],["429","1155","1155","525","105","7"],["1430","4576","5720","3520","1100","160","8"],["4862","18018","27027","21021","9009","2079","231","9"],["16796","70720","123760","116480","63700","20384","3640","320","10"],["58786","277134","554268","612612","408408","168168","42042","6006","429","11"],["208012","1085280","2441880","3100800","2441880","1233792","399840","80640","9450","560","12"],["742900","4249388","10623470","15242370","13856700","8314020","3325608","875160","145860","14300","715","13"],["2674440","16640960","45762640","73220224","75508356","52527552","25069968","8186112","1790712","251328","20944","896","14"],["9694845","65189475","195568425","345262775","398380125","315517059","175287255","68590665","18706545","3464175","415701","29835","1105","15"],["35357670","255487680","830334960","1603405440","2047205160","1819737920","1154833680","527923968","173225052","40168128","6390384","663936","41496","1344","16"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","8","34","152","706","3376","16514","82256","415810","2127864","11002146","57390504","301654146","1596108000","8494776834"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["3"],["4"],["1"]]},"terms":[{"coeff":{"coeffs":[["6","3"],["10","6"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-3"],["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174650,"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-ca102fc79ed966ef"]},{"id":174651,"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-ca102fc79ed966ef"]},{"id":174652,"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-ca102fc79ed966ef"]},{"id":174653,"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-ca102fc79ed966ef"]}],"region_exact":{},"incoming_edges":[{"id":"e-ca102fc79ed966ef","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-8cac8e8480894b25","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":"eff14e70835e8ea33f377b1f145fabb5f732533bd400554df865acceb9bd9154","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-c7016e67374634ec","role":"sequence","name":"reversal of Triangle of Dyck paths counted by number of long interior inclines.","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-0dca9696e98a68cd","kind":"fingerprint_related","node_a_id":"n-8cac8e8480894b25","node_b_id":"n-3a81c23f7a892ffd","fingerprint":"f6-5cd009bfc7bc0bb1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5cd009bfc7bc0bb1"},"other_id":"n-3a81c23f7a892ffd","other":{"id":"n-3a81c23f7a892ffd","name":"mobius_x_over_x_plus_one of reversal of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["8"],["18","30"],["32","128","112"]],"card_rows":[["8"],["18","30"],["32","128","112"],["50","350","700","420"]],"row_sums_preview":["8","48","272","1520","8472","47264","264224","1480608"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7a4827a398e28fe4","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-8b10d4d15ed56acc","e-14c2f28924cda6a6"],"text":"A352880 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A352880 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-23f9fc4dcd509c13","kind":"fingerprint_related","node_a_id":"n-d54c958727b939eb","node_b_id":"n-8cac8e8480894b25","fingerprint":"f6-5cd009bfc7bc0bb1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5cd009bfc7bc0bb1"},"other_id":"n-d54c958727b939eb","other":{"id":"n-d54c958727b939eb","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["8"],["30","18"],["112","128","32"]],"card_rows":[["8"],["30","18"],["112","128","32"],["420","700","350","50"]],"row_sums_preview":["8","48","272","1520","8472","47264","264224","1480608"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7a4827a398e28fe4","steps":["x \u21a6 x+1"],"edges":["e-5e1fa9e41a7375d8"],"text":"A352880 \u2192 x \u21a6 x+1"},"headline":"A352880 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-3c452f4a6b530d2c","kind":"fingerprint_related","node_a_id":"n-f7cfaa6a3770b14b","node_b_id":"n-8cac8e8480894b25","fingerprint":"f6-5cd009bfc7bc0bb1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5cd009bfc7bc0bb1"},"other_id":"n-f7cfaa6a3770b14b","other":{"id":"n-f7cfaa6a3770b14b","name":"mobius_x_over_x_plus_one of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["3","5"],["4","16","14"]],"card_rows":[["2"],["3","5"],["4","16","14"],["5","35","70","42"]],"row_sums_preview":["2","8","34","152","706","3376","16514","82256"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["x \u21a6 x/(x+1)"],"edges":["e-c9da786527cfa7e3"],"text":"A108838 \u2192 x \u21a6 x/(x+1)"},"headline":"A108838 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-4c43907ae7a33e56","kind":"fingerprint_related","node_a_id":"n-8cac8e8480894b25","node_b_id":"n-fabe4ac480a936a7","fingerprint":"f6-5cd009bfc7bc0bb1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5cd009bfc7bc0bb1"},"other_id":"n-fabe4ac480a936a7","other":{"id":"n-fabe4ac480a936a7","name":"shift_x_plus_one of derivative of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["3"],["20","12"]],"card_rows":[["3"],["20","12"],["105","120","30"],["504","840","420","60"]],"row_sums_preview":["0","3","32","255","1824","12355","81024","520191"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5aa52e3acf29a627","steps":["D"],"edges":["e-e2cb16f67cf9b10f"],"text":"A345013 \u2192 D"},"headline":"A345013 \u2192 D","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:     2;\n  1:     5,      3;\n  2:    14,     16,      4;\n  3:    42,     70,     35,      5;\n  4:   132,    288,    216,     64,      6;\n  5:   429,   1155,   1155,    525,    105,      7;\n  6:  1430,   4576,   5720,   3520,   1100,    160,     8;\n  7:  4862,  18018,  27027,  21021,   9009,   2079,   231,    9;\n  8: 16796,  70720, 123760, 116480,  63700,  20384,  3640,  320,  10;\n  9: 58786, 277134, 554268, 612612, 408408, 168168, 42042, 6006, 429, 11;\n","recurrence_sentence":"The row polynomials satisfy (n^2+4*n+3)*P(n,x) = (4*n^2+10*n+6+(3*n^2+6*n+3)*x)*P(n-1,x) - ((2*n+3)*x+(2*n+3)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.","mathematica":"P[0, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 10*n + 6 + (3*n^2 + 6*n + 3)*x)*P[n - 1, x] - ((2*n + 3)*x + (2*n + 3)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 10*n + 6 + (3*n^2 + 6*n + 3)*x)*v[i - 1] - ((2*n + 3)*x + (2*n + 3)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n + 3)); 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-c963abfc0266d66c","steps":["reverse","x \u21a6 x+1"],"edges":["e-7ac651410315272d","e-ca102fc79ed966ef"],"text":"A108838 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A108838 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}
