{"id":"n-d6f28c09f7011372","revision":0,"first_index":0,"fingerprint":"f6-c880512e8efe320e","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["0","1","3"],["0","0","1","8","13"],["0","0","0","1","15","57","63"],["0","0","0","0","1","24","156","384","321"],["0","0","0","0","0","1","35","340","1380","2505","1683"],["0","0","0","0","0","0","1","48","645","3800","11145","16008","8989"],["0","0","0","0","0","0","0","1","63","1113","8855","37065","84903","100849","48639"],["0","0","0","0","0","0","0","0","1","80","1792","18368","102340","330848","621040","628736","265729"],["0","0","0","0","0","0","0","0","0","1","99","2736","34944","247716","1055628","2774352","4408848","3888657","1462563"],["0","0","0","0","0","0","0","0","0","0","1","120","4005","62160","543060","2919168","10041780","22209120","30589065","23900040","8097453"],["0","0","0","0","0","0","0","0","0","0","0","1","143","5665","104775","1101540","7241388","31222884","89899260","171521625","208411335","146146473","45046719"],["0","0","0","0","0","0","0","0","0","0","0","0","1","168","7788","168960","2098305","16484688","86427264","310394304","767708865","1287357720","1399199868","889928064","251595969"],["0","0","0","0","0","0","0","0","0","0","0","0","0","1","195","10452","262548","3794505","34992243","218103600","948220416","2913982929","6312758595","9439841268","9279936900","5399971161","1409933619"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","1","224","13741","395304","6567561","70078008","510266757","2626557648","9736383657","26124706608","50326289013","67892966232","60919688193","32668236552","7923848253"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","4","22","136","886","5944","40636","281488","1968934","13875544","98365972","700701808","5011371964","35961808432","258805997752"],"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","0"],["1","3","3","1"]]},"terms":[{"coeff":{"coeffs":[["0","2","14","38","50","32","8"],["0","-1","-8","-26","-40","-29","-8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","2","12","28","32","18","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178260,"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-48cb9c8ffb85d87e"]},{"id":178261,"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-48cb9c8ffb85d87e"]},{"id":178262,"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-48cb9c8ffb85d87e"]}],"region_exact":{},"incoming_edges":[{"id":"e-48cb9c8ffb85d87e","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\":2}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-d6f28c09f7011372","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":"bd3f93a7ec5586102efef8627df2b1fb47a6d18646da3613c183e7f94ae2a8d0","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-e4d248920a317259","role":"sequence","name":"Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","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-1c86f30da878a74a","kind":"fingerprint_related","node_a_id":"n-d6f28c09f7011372","node_b_id":"n-c73008ca3f0e3c9c","fingerprint":"f6-c880512e8efe320e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c880512e8efe320e"},"other_id":"n-c73008ca3f0e3c9c","other":{"id":"n-c73008ca3f0e3c9c","name":"shift_x_plus_one of basis_map_A007318 of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","first_index":0,"fingerprint":"f6-c880512e8efe320e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","2"],["78","48","6"]],"card_rows":[["1"],["6","2"],["78","48","6"],["1260","1140","300","20"]],"row_sums_preview":["1","8","132","2720","62020","1497888","37547664","966066816"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fdc03fad975441d0","steps":["basis A007318","x \u21a6 x+1"],"edges":["e-2cd9cd27e716475b","e-8429eaba0caeabae"],"text":"A069466 \u2192 basis A007318 \u2192 x \u21a6 x+1"},"headline":"A069466 \u2192 basis A007318 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-1e9280f214f8ee48","kind":"fingerprint_related","node_a_id":"n-118f91023ad73cd8","node_b_id":"n-d6f28c09f7011372","fingerprint":"f6-c880512e8efe320e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c880512e8efe320e"},"other_id":"n-118f91023ad73cd8","other":{"id":"n-118f91023ad73cd8","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","first_index":1,"fingerprint":"f6-c880512e8efe320e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["13","8","1"]],"card_rows":[["1"],["3","1"],["13","8","1"],["63","57","15","1"]],"row_sums_preview":["1","4","22","136","886","5944","40636","281488"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-78ba05b4a3dd925d","steps":["basis A007318"],"edges":["e-32321abfdad5764e"],"text":"A104684 \u2192 basis A007318"},"headline":"A104684 \u2192 basis A007318","chain":null}},{"id":"l-665e58ac028ccd5e","kind":"fingerprint_related","node_a_id":"n-c8c2604dc29a0657","node_b_id":"n-d6f28c09f7011372","fingerprint":"f6-c880512e8efe320e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c880512e8efe320e"},"other_id":"n-c8c2604dc29a0657","other":{"id":"n-c8c2604dc29a0657","name":"shift_x_plus_one of basis_map_A007318 of Square the entries of Pascal's triangle.","first_index":0,"fingerprint":"f6-c880512e8efe320e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["13","8","1"]],"card_rows":[["1"],["3","1"],["13","8","1"],["63","57","15","1"]],"row_sums_preview":["1","4","22","136","886","5944","40636","281488"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b13531ad79e7c629","steps":["x \u21a6 x+1","basis A007318"],"edges":["e-041c07f9bf4043d7","e-9ff4db41417b157a"],"text":"A008459 \u2192 x \u21a6 x+1 \u2192 basis A007318"},"headline":"A008459 \u2192 x \u21a6 x+1 \u2192 basis A007318","chain":null}},{"id":"l-6db8636b42b0210e","kind":"fingerprint_related","node_a_id":"n-d6f28c09f7011372","node_b_id":"n-7954740262dbc1d9","fingerprint":"f6-c880512e8efe320e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c880512e8efe320e"},"other_id":"n-7954740262dbc1d9","other":{"id":"n-7954740262dbc1d9","name":"reversal of basis_map_A007318 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","first_index":1,"fingerprint":"f6-c880512e8efe320e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","8","13"]],"card_rows":[["1"],["1","3"],["1","8","13"],["1","15","57","63"]],"row_sums_preview":["1","4","22","136","886","5944","40636","281488"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-78ba05b4a3dd925d","steps":["basis A007318","reverse"],"edges":["e-32321abfdad5764e","e-fb45aeb07dfec5df"],"text":"A104684 \u2192 basis A007318 \u2192 reverse"},"headline":"A104684 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-e37b10521089fb4e","kind":"fingerprint_related","node_a_id":"n-939b76c3ba274588","node_b_id":"n-d6f28c09f7011372","fingerprint":"f6-c880512e8efe320e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c880512e8efe320e"},"other_id":"n-939b76c3ba274588","other":{"id":"n-939b76c3ba274588","name":"mobius_x_over_x_plus_one of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-c880512e8efe320e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","8","13"]],"card_rows":[["1"],["1","3"],["1","8","13"],["1","15","57","63"]],"row_sums_preview":["1","4","22","136","886","5944","40636","281488"],"regions":["[-1, 0)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4689fb1386d440ad","steps":["x \u21a6 x/(x+1)"],"edges":["e-5184591996606eba"],"text":"A063007 \u2192 x \u21a6 x/(x+1)"},"headline":"A063007 \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: 0, 1, 3;\n  2: 0, 0, 1, 8, 13;\n  3: 0, 0, 0, 1, 15, 57,  63;\n  4: 0, 0, 0, 0,  1, 24, 156, 384,  321;\n  5: 0, 0, 0, 0,  0,  1,  35, 340, 1380, 2505,  1683;\n  6: 0, 0, 0, 0,  0,  0,   1,  48,  645, 3800, 11145, 16008,   8989;\n  7: 0, 0, 0, 0,  0,  0,   0,   1,   63, 1113,  8855, 37065,  84903, 100849,   48639;\n  8: 0, 0, 0, 0,  0,  0,   0,   0,    1,   80,  1792, 18368, 102340, 330848,  621040,  628736,  265729;\n  9: 0, 0, 0, 0,  0,  0,   0,   0,    0,    1,    99,  2736,  34944, 247716, 1055628, 2774352, 4408848, 3888657, 1462563;\n","recurrence_sentence":"The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = -((n-2)*x+(8*n-14)*x^2+(26*n-38)*x^3+(40*n-50)*x^4+(29*n-32)*x^5+(8*n-8)*x^6)*P(n-1,x) + (2*x^2+12*x^3+28*x^4+32*x^5+18*x^6+4*x^7)*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)*x + (8*n - 14)*x^2 + (26*n - 38)*x^3 + (40*n - 50)*x^4 + (29*n - 32)*x^5 + (8*n - 8)*x^6)*P[n - 1, x] + (2*x^2 + 12*x^3 + 28*x^4 + 32*x^5 + 18*x^6 + 4*x^7)*D[P[n - 1, x], x])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];\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)*x + (8*n - 14)*x^2 + (26*n - 38)*x^3 + (40*n - 50)*x^4 + (29*n - 32)*x^5 + (8*n - 8)*x^6)*v[i - 1] + (2*x^2 + 12*x^3 + 28*x^4 + 32*x^5 + 18*x^6 + 4*x^7)*deriv(v[i - 1], x))/(n + 3*n*x + 3*n*x^2 + n*x^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-e4d248920a317259","steps":["x \u21a6 x/(x+1)"],"edges":["e-48cb9c8ffb85d87e"],"text":"A109983 \u2192 x \u21a6 x/(x+1)"},"headline":"A109983 \u2192 x \u21a6 x/(x+1)","chain":null}