{"id":"n-b4a70761c59dc931","revision":1,"first_index":0,"fingerprint":"f6-98cc1557e904900a","name":"mobius_x_over_x_plus_one of Triangle read by rows: triangle of repeated stepped binomial coefficients.","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":true,"rows":[["1"],["0","1"],["0","1","2"],["0","0","1","2"],["0","0","1","4","4"],["0","0","0","1","4","4"],["0","0","0","1","6","12","8"],["0","0","0","0","1","6","12","8"],["0","0","0","0","1","8","24","32","16"],["0","0","0","0","0","1","8","24","32","16"],["0","0","0","0","0","1","10","40","80","80","32"],["0","0","0","0","0","0","1","10","40","80","80","32"],["0","0","0","0","0","0","1","12","60","160","240","192","64"],["0","0","0","0","0","0","0","1","12","60","160","240","192","64"],["0","0","0","0","0","0","0","1","14","84","280","560","672","448","128"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","3","3","9","9","27","27","81","81","243","243","729","729","2187"],"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":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["1","8","17","14","4"],["-1","-7","-15","-13","-4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","10","18","14","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160735,"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-3c0198f97249d373"]},{"id":160736,"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-3c0198f97249d373"]},{"id":160737,"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-3c0198f97249d373"]}],"region_exact":{},"incoming_edges":[{"id":"e-3c0198f97249d373","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-b4a70761c59dc931","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":"e38d4335a4e9fdb25026d0159219f9ee4c7426922c46a7f674e6ffecb8ad597a","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-35bd700768cac55f","role":"sequence","name":"Triangle read by rows: triangle of repeated stepped binomial coefficients.","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-0a3fdd8edfcf7b62","kind":"fingerprint_related","node_a_id":"n-b4a70761c59dc931","node_b_id":"n-98c21bd2a6c7c1d4","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-98c21bd2a6c7c1d4","other":{"id":"n-98c21bd2a6c7c1d4","name":"basis_map_A013611 of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["4","8"]],"card_rows":[["1"],["2"],["4","8"],["8","16"]],"row_sums_preview":["1","2","12","24","144","288","1728","3456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b6aca576a689e9f0","steps":["basis A013611"],"edges":["e-6f6524a2909f4dfb"],"text":"A274496 \u2192 basis A013611"},"headline":"A274496 \u2192 basis A013611","chain":null}},{"id":"l-3b6d91f6db7b6417","kind":"fingerprint_related","node_a_id":"n-b4a70761c59dc931","node_b_id":"n-8c8ff2e102336282","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-8c8ff2e102336282","other":{"id":"n-8c8ff2e102336282","name":"shift_x_plus_one of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["2","1"]],"row_sums_preview":["1","1","3","3","9","9","27","27"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["x \u21a6 x+1"],"edges":["e-009b68c8336dc2e0"],"text":"A152198 \u2192 x \u21a6 x+1"},"headline":"A152198 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-3ced7d90f21861a6","kind":"fingerprint_related","node_a_id":"n-7c641d81e21514ce","node_b_id":"n-b4a70761c59dc931","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-7c641d81e21514ce","other":{"id":"n-7c641d81e21514ce","name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3"],["3","6"]],"card_rows":[["1"],["3"],["3","6"],["9","18"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A274498"],"derivation":{"root":"n-7c641d81e21514ce","steps":[],"edges":[],"text":"A274498"},"headline":"A274498: Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","chain":null}},{"id":"l-460fce14899dff3d","kind":"fingerprint_related","node_a_id":"n-b4a70761c59dc931","node_b_id":"n-b3c0d8fe47ca78c3","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-b3c0d8fe47ca78c3","other":{"id":"n-b3c0d8fe47ca78c3","name":"derivative of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["2"],["4","8"],["4","8"]],"row_sums_preview":["0","0","2","2","12","12","54","54"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba3bb18490b15e2e","steps":["D"],"edges":["e-107b5f015c60ec70"],"text":"A188440 \u2192 D"},"headline":"A188440 \u2192 D","chain":null}},{"id":"l-5634e67d27f78d14","kind":"fingerprint_related","node_a_id":"n-b4a70761c59dc931","node_b_id":"n-2b030ef584988651","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-2b030ef584988651","other":{"id":"n-2b030ef584988651","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3"],["9","18"]],"card_rows":[["1"],["3"],["9","18"],["27","54"]],"row_sums_preview":["1","3","27","81","729","2187","19683","59049"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c641d81e21514ce","steps":["basis A013610"],"edges":["e-b7097430306b08af"],"text":"A274498 \u2192 basis A013610"},"headline":"A274498 \u2192 basis A013610","chain":null}},{"id":"l-614d936efd014750","kind":"fingerprint_related","node_a_id":"n-b4a70761c59dc931","node_b_id":"n-4d941e7b1ccdb34d","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-4d941e7b1ccdb34d","other":{"id":"n-4d941e7b1ccdb34d","name":"basis_map_A013611 of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["2","4"]],"card_rows":[["1"],["1"],["2","4"],["2","4"]],"row_sums_preview":["1","1","6","6","36","36","216","216"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["basis A013611"],"edges":["e-0bf2c6340da57ed0"],"text":"A152198 \u2192 basis A013611"},"headline":"A152198 \u2192 basis A013611","chain":null}},{"id":"l-682caa08a65728af","kind":"fingerprint_related","node_a_id":"n-b4a70761c59dc931","node_b_id":"n-7619704d59388f48","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-7619704d59388f48","other":{"id":"n-7619704d59388f48","name":"basis_map_A013610 of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["3","6"]],"card_rows":[["1"],["1"],["3","6"],["3","6"]],"row_sums_preview":["1","1","9","9","81","81","729","729"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba3bb18490b15e2e","steps":["basis A013610"],"edges":["e-e7473649d9984108"],"text":"A188440 \u2192 basis A013610"},"headline":"A188440 \u2192 basis A013610","chain":null}},{"id":"l-6b459278936520f6","kind":"fingerprint_related","node_a_id":"n-b4a70761c59dc931","node_b_id":"n-f42b095ef0d54419","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-f42b095ef0d54419","other":{"id":"n-f42b095ef0d54419","name":"derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["6"]],"card_rows":[["6"],["18"],["36","72"],["108","216"]],"row_sums_preview":["0","0","6","18","108","324","1458","4374"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c641d81e21514ce","steps":["D"],"edges":["e-7b8439e885873d7d"],"text":"A274498 \u2192 D"},"headline":"A274498 \u2192 D","chain":null}},{"id":"l-73b0b35f2f252e71","kind":"fingerprint_related","node_a_id":"n-b4a70761c59dc931","node_b_id":"n-d321ff0d3bb39024","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-d321ff0d3bb39024","other":{"id":"n-d321ff0d3bb39024","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["4","2"]],"card_rows":[["1"],["2"],["4","2"],["8","4"]],"row_sums_preview":["1","2","6","12","36","72","216","432"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b6aca576a689e9f0","steps":["x \u21a6 x+1"],"edges":["e-5428ec8f88838d6b"],"text":"A274496 \u2192 x \u21a6 x+1"},"headline":"A274496 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-e28c438a450047de","kind":"fingerprint_related","node_a_id":"n-ba3bb18490b15e2e","node_b_id":"n-b4a70761c59dc931","fingerprint":"f6-98cc1557e904900a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-98cc1557e904900a"},"other_id":"n-ba3bb18490b15e2e","other":{"id":"n-ba3bb18490b15e2e","name":"Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","first_index":0,"fingerprint":"f6-98cc1557e904900a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["1","2"]],"card_rows":[["1"],["1"],["1","2"],["1","2"]],"row_sums_preview":["1","1","3","3","9","9","27","27"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A188440"],"derivation":{"root":"n-ba3bb18490b15e2e","steps":[],"edges":[],"text":"A188440"},"headline":"A188440: Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","chain":null}},{"id":"l-0dfb99d92c506bd2","kind":"oeis_match","node_a_id":"n-b4a70761c59dc931","node_b_id":null,"fingerprint":null,"oeis_id":"A188440","in_catalog":1,"target":"triangle","row_alignment":"node_lengths","rows_matched":15,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":true,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":15,"target":"triangle","terms_compared":64},"in_catalog":true,"oeis_id":"A188440","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-30d1cfb18e083863","kind":"oeis_match","node_a_id":"n-b4a70761c59dc931","node_b_id":null,"fingerprint":null,"oeis_id":"A162436","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A162436","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-61f0022970fbaadb","kind":"oeis_match","node_a_id":"n-b4a70761c59dc931","node_b_id":null,"fingerprint":null,"oeis_id":"A108411","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":"A108411","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-da80db81a90e4b06","kind":"oeis_match","node_a_id":"n-b4a70761c59dc931","node_b_id":null,"fingerprint":null,"oeis_id":"A056449","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A056449","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A056449","relation":"match_row_sums"},{"oeis_id":"A108411","relation":"match_row_sums"},{"oeis_id":"A162436","relation":"match_row_sums"},{"oeis_id":"A188440","relation":"match_triangle"}],"bfiles":[{"node_id":"n-b4a70761c59dc931","oeis_id":"A056449","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":38,"oeis_terms_available":38,"term_count":200,"raw_size":5759},{"node_id":"n-b4a70761c59dc931","oeis_id":"A108411","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":39,"oeis_terms_available":39,"term_count":200,"raw_size":5712},{"node_id":"n-b4a70761c59dc931","oeis_id":"A162436","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":38,"oeis_terms_available":38,"term_count":200,"raw_size":5761},{"node_id":"n-b4a70761c59dc931","oeis_id":"A188440","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":62,"terms_matched":64,"oeis_terms_agreeing":68,"oeis_terms_available":68,"term_count":992,"raw_size":11892}],"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;\n  2: 0, 1, 2;\n  3: 0, 0, 1, 2;\n  4: 0, 0, 1, 4, 4;\n  5: 0, 0, 0, 1, 4,  4;\n  6: 0, 0, 0, 1, 6, 12,  8;\n  7: 0, 0, 0, 0, 1,  6, 12,  8;\n  8: 0, 0, 0, 0, 1,  8, 24, 32, 16;\n  9: 0, 0, 0, 0, 0,  1,  8, 24, 32, 16;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(n-1+(7*n-8)*x+(15*n-17)*x^2+(13*n-14)*x^3+(4*n-4)*x^4)*P(n-1,x) + (2*x+10*x^2+18*x^3+14*x^4+4*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 - 1 + (7*n - 8)*x + (15*n - 17)*x^2 + (13*n - 14)*x^3 + (4*n - 4)*x^4)*P[n - 1, x] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(1 + 2*x + 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 - 1 + (7*n - 8)*x + (15*n - 17)*x^2 + (13*n - 14)*x^3 + (4*n - 4)*x^4)*v[i - 1] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(1 + 2*x + 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}},{"oeis_id":"A188440","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"low","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1;\n  2: 1, 2;\n  3: 1, 2;\n  4: 1, 4,  4;\n  5: 1, 4,  4;\n  6: 1, 6, 12,  8;\n  7: 1, 6, 12,  8;\n  8: 1, 8, 24, 32, 16;\n  9: 1, 8, 24, 32, 16;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where (1+2*x+x^2)*P(n,x) = -(n-1+(7*n-8)*x+(15*n-17)*x^2+(13*n-14)*x^3+(4*n-4)*x^4)*P(n-1,x) + (2*x+10*x^2+18*x^3+14*x^4+4*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 - 1 + (7*n - 8)*x + (15*n - 17)*x^2 + (13*n - 14)*x^3 + (4*n - 4)*x^4)*P[n - 1, x] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];\nTable[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], 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 - 1 + (7*n - 8)*x + (15*n - 17)*x^2 + (13*n - 14)*x^3 + (4*n - 4)*x^4)*v[i - 1] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":"low","regular_triangle":false}}],"derivation":{"root":"n-35bd700768cac55f","steps":["x \u21a6 x/(x+1)"],"edges":["e-3c0198f97249d373"],"text":"A103633 \u2192 x \u21a6 x/(x+1)"},"headline":"A103633 \u2192 x \u21a6 x/(x+1)","chain":null}