{"id":"n-972e33de68e48c50","revision":0,"first_index":1,"fingerprint":"f6-5870c1c52d074a69","name":"shift_x_plus_one of reversal of Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["5","5","1"],["13","20","9","1"],["34","71","51","14","1"],["89","235","233","105","20","1"],["233","744","942","594","190","27","1"],["610","2285","3522","2860","1295","315","35","1"],["1597","6865","12473","12402","7285","2534","490","44","1"],["4181","20284","42447","49963","36122","16407","4578","726","54","1"],["10946","59155","140109","190570","163730","91959","33705","7776","1035","65","1"],["28657","170711","451441","696787","693835","464723","211519","64449","12573","1430","77","1"],["75025","488400","1426380","2463300","2790100","2171850","1182650","449240","116325","19525","1925","90","1"],["196418","1387225","4434420","8472280","10758050","9546570","6056850","2760810","894465","200200","29315","2535","104","1"],["514229","3916061","13599505","28481220","40075630","39940460","28947240","15409740","6007560","1688115","331045","42770","3276","119","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","11","43","171","683","2731","10923","43691","174763","699051","2796203","11184811","44739243","178956971"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["-14","-11"],["8","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-5","-9","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172495,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-32d8bfc6e7ec42bf"]},{"id":172496,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-32d8bfc6e7ec42bf"]},{"id":172497,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-32d8bfc6e7ec42bf"]}],"region_exact":{},"incoming_edges":[{"id":"e-32d8bfc6e7ec42bf","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-972e33de68e48c50","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":"84d0bb1b744b60433f80b05096ad830b7768615c5233133bf169a935b57a4b21","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-d8cfb1e39c5d56b0","role":"sequence","name":"reversal of Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-1e98cf8f4a3d64e9","kind":"fingerprint_related","node_a_id":"n-fac4d4a000d744e8","node_b_id":"n-972e33de68e48c50","fingerprint":"f6-5870c1c52d074a69","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5870c1c52d074a69"},"other_id":"n-fac4d4a000d744e8","other":{"id":"n-fac4d4a000d744e8","name":"Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).","first_index":0,"fingerprint":"f6-5870c1c52d074a69","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["5","5","1"]],"card_rows":[["1"],["2","1"],["5","5","1"],["13","20","9","1"]],"row_sums_preview":["1","3","11","43","171","683","2731","10923"],"regions":["(-inf, +inf)","(-inf, -1]","(-inf, 0]","(-inf, 0)"],"region_exact":{},"oeis_ids":["A060920"],"derivation":{"root":"n-fac4d4a000d744e8","steps":[],"edges":[],"text":"A060920"},"headline":"A060920: Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).","chain":null}},{"id":"l-9f4415c5d36bd407","kind":"fingerprint_related","node_a_id":"n-aa424fc3fffbd09d","node_b_id":"n-972e33de68e48c50","fingerprint":"f6-5870c1c52d074a69","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5870c1c52d074a69"},"other_id":"n-aa424fc3fffbd09d","other":{"id":"n-aa424fc3fffbd09d","name":"derivative of Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.","first_index":0,"fingerprint":"f6-5870c1c52d074a69","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["8","4"]],"card_rows":[["2"],["8","4"],["30","30","6"],["104","160","72","8"]],"row_sums_preview":["0","2","12","66","344","1710","8196","38234"],"regions":["(-inf, +inf)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9b40f045d129e237","steps":["D"],"edges":["e-fa6dba774ff7b5f5"],"text":"A122076 \u2192 D"},"headline":"A122076 \u2192 D","chain":null}},{"id":"l-a2d61a4744adc653","kind":"fingerprint_related","node_a_id":"n-80f3c43c239c22ff","node_b_id":"n-972e33de68e48c50","fingerprint":"f6-5870c1c52d074a69","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5870c1c52d074a69"},"other_id":"n-80f3c43c239c22ff","other":{"id":"n-80f3c43c239c22ff","name":"reversal of Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).","first_index":0,"fingerprint":"f6-5870c1c52d074a69","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","5"]],"card_rows":[["1"],["1","2"],["1","5","5"],["1","9","20","13"]],"row_sums_preview":["1","3","11","43","171","683","2731","10923"],"regions":["(-inf, +inf)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fac4d4a000d744e8","steps":["reverse"],"edges":["e-0c6cc2be731e428d"],"text":"A060920 \u2192 reverse"},"headline":"A060920 \u2192 reverse","chain":null}},{"id":"l-b658b9672aaf4f3f","kind":"fingerprint_related","node_a_id":"n-c0364af117c168ab","node_b_id":"n-972e33de68e48c50","fingerprint":"f6-5870c1c52d074a69","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5870c1c52d074a69"},"other_id":"n-c0364af117c168ab","other":{"id":"n-c0364af117c168ab","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","first_index":1,"fingerprint":"f6-5870c1c52d074a69","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","5"]],"card_rows":[["1"],["1","2"],["1","5","5"],["1","9","20","13"]],"row_sums_preview":["1","3","11","43","171","683","2731","10923"],"regions":["[-1, 0)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0eba401b117f16a9","steps":["x \u21a6 x/(x+1)"],"edges":["e-734a74afb69f9916"],"text":"A085478 \u2192 x \u21a6 x/(x+1)"},"headline":"A085478 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-fb6c0f5bb51595e4","kind":"fingerprint_related","node_a_id":"n-972e33de68e48c50","node_b_id":"n-208323846157519b","fingerprint":"f6-5870c1c52d074a69","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5870c1c52d074a69"},"other_id":"n-208323846157519b","other":{"id":"n-208323846157519b","name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","first_index":0,"fingerprint":"f6-5870c1c52d074a69","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["8","4"],["80","80","16"]],"card_rows":[["1"],["8","4"],["80","80","16"],["832","1280","576","64"]],"row_sums_preview":["1","12","176","2752","43776","699392","11186176","178962432"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-36be6b9e5c42de00","e-6bf2133ea5312077"],"text":"A103327 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A103327 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    2,     1;\n   3:    5,     5,     1;\n   4:   13,    20,     9,     1;\n   5:   34,    71,    51,    14,     1;\n   6:   89,   235,   233,   105,    20,     1;\n   7:  233,   744,   942,   594,   190,    27,    1;\n   8:  610,  2285,  3522,  2860,  1295,   315,   35,   1;\n   9: 1597,  6865, 12473, 12402,  7285,  2534,  490,  44,  1;\n  10: 4181, 20284, 42447, 49963, 36122, 16407, 4578, 726, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = (8*n-14+(6*n-11)*x)*P(n-1,x) - (5+9*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n - 14 + (6*n - 11)*x)*P[n - 1, x] - (5 + 9*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n - 14 + (6*n - 11)*x)*v[i - 1] - (5 + 9*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-0eba401b117f16a9","steps":["reverse","x \u21a6 x+1"],"edges":["e-6116c33129f68e6c","e-32d8bfc6e7ec42bf"],"text":"A085478 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A085478 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}