{"id":"n-7bfbd1a79b2d2b76","revision":1,"first_index":0,"fingerprint":"f6-3e6ccf881df1d3b2","name":"Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0011 (n,k>=0).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["4"],["8"],["15","1"],["28","4"],["52","12"],["96","32"],["177","78","1"],["326","180","6"],["600","400","24"],["1104","864","80"],["2031","1827","237","1"],["3736","3800","648","8"],["6872","7800","1672","40"]],"degrees":[0,0,0,0,1,1,1,1,2,2,2,2,3,3,3],"row_sums":["1","2","4","8","16","32","64","128","256","512","1024","2048","4096","8192","16384"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A118884","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"],["4"],["8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","1"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160285,"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-40a62e5a79cc0439"]}],"region_exact":{},"incoming_edges":[{"id":"e-40a62e5a79cc0439","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1/48\",\"intercept\":\"-1/48\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-7bfbd1a79b2d2b76","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":"041175fe2dd64904370c44594eefa8d7e9b7f741548af90088514983b4951a45","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"1"}}}],"premises":[{"position":0,"node_id":"n-a2db6b6ded7a955c","role":"sequence","name":"Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.","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-1ac862603b5aea74","kind":"oeis_match","node_a_id":"n-7bfbd1a79b2d2b76","node_b_id":null,"fingerprint":null,"oeis_id":"A166444","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A166444","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-1f55bc7308220836","kind":"oeis_match","node_a_id":"n-7bfbd1a79b2d2b76","node_b_id":null,"fingerprint":null,"oeis_id":"A118884","in_catalog":1,"target":"triangle","row_alignment":"node_lengths","rows_matched":15,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":15,"target":"triangle","terms_compared":36},"in_catalog":true,"oeis_id":"A118884","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-25ca56d42a78c12d","kind":"oeis_match","node_a_id":"n-7bfbd1a79b2d2b76","node_b_id":null,"fingerprint":null,"oeis_id":"A011782","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A011782","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-599ae2c815f9d98a","kind":"oeis_match","node_a_id":"n-7bfbd1a79b2d2b76","node_b_id":null,"fingerprint":null,"oeis_id":"A131577","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A131577","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c03b240027ee8fdf","kind":"oeis_match","node_a_id":"n-7bfbd1a79b2d2b76","node_b_id":null,"fingerprint":null,"oeis_id":"A220051","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":"A220051","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ca41215fb844c70c","kind":"oeis_match","node_a_id":"n-7bfbd1a79b2d2b76","node_b_id":null,"fingerprint":null,"oeis_id":"A118655","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":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A118655","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-cd0d3d48879b689c","kind":"oeis_match","node_a_id":"n-7bfbd1a79b2d2b76","node_b_id":null,"fingerprint":null,"oeis_id":"A034008","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A034008","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ebbd00f242e1c06b","kind":"oeis_match","node_a_id":"n-7bfbd1a79b2d2b76","node_b_id":null,"fingerprint":null,"oeis_id":"A050732","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":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A050732","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fb8391dd734987a9","kind":"oeis_match","node_a_id":"n-7bfbd1a79b2d2b76","node_b_id":null,"fingerprint":null,"oeis_id":"A000079","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":"A000079","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A118884","relation":"identification"},{"oeis_id":"A000079","relation":"match_row_sums"},{"oeis_id":"A011782","relation":"match_row_sums"},{"oeis_id":"A034008","relation":"match_row_sums"},{"oeis_id":"A050732","relation":"match_row_sums"},{"oeis_id":"A118655","relation":"match_row_sums"},{"oeis_id":"A131577","relation":"match_row_sums"},{"oeis_id":"A166444","relation":"match_row_sums"},{"oeis_id":"A220051","relation":"match_row_sums"},{"oeis_id":"A118884","relation":"match_triangle"}],"bfiles":[{"node_id":"n-7bfbd1a79b2d2b76","oeis_id":"A000079","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":35,"oeis_terms_available":35,"term_count":200,"raw_size":6981},{"node_id":"n-7bfbd1a79b2d2b76","oeis_id":"A118884","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":87,"terms_matched":36,"oeis_terms_agreeing":52,"oeis_terms_available":52,"term_count":990,"raw_size":18176}],"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:   2;\n  2:   4;\n  3:   8;\n  4:  15,   1;\n  5:  28,   4;\n  6:  52,  12;\n  7:  96,  32;\n  8: 177,  78, 1;\n  9: 326, 180, 6;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-x)*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 4, P(3,x) = 8.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 4;\nP[3, x_] := 8;\nP[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 4, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 4; v[4] = 8; for(n = 4, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - x)*v[i - 4]); 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":"A118884","target":"triangle","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:   2;\n  2:   4;\n  3:   8;\n  4:  15,   1;\n  5:  28,   4;\n  6:  52,  12;\n  7:  96,  32;\n  8: 177,  78, 1;\n  9: 326, 180, 6;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-x)*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 4, P(3,x) = 8.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 4;\nP[3, x_] := 8;\nP[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 4, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 4; v[4] = 8; for(n = 4, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - x)*v[i - 4]); 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-a2db6b6ded7a955c","steps":["x \u21a6 (1/48)x\u22121/48"],"edges":["e-40a62e5a79cc0439"],"text":"A318772 \u2192 x \u21a6 (1/48)x\u22121/48"},"headline":"A118884: Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0011 (n,k>=0).","chain":"A318772 \u2192 x \u21a6 (1/48)x\u22121/48"}
