{"id":"n-60c7f085d1907563","revision":1,"first_index":1,"fingerprint":"f6-e7d8403424fc1d9b","name":"basis_map_A013609 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.","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"],["3","2"],["13","16","4"],["63","114","60","8"],["321","768","624","192","16"],["1683","5010","5520","2720","560","32"],["8989","32016","44580","30400","10320","1536","64"],["48639","201698","339612","296520","141680","35616","4032","128"],["265729","1257472","2484160","2646784","1637440","587776","114688","10240","256"],["1462563","7777314","17635392","22194816","16890048","7926912","2236416","350208","25344","512"],["8097453","47800080","122356260","177672960","160668480","93413376","34755840","7956480","1025280","61440","1024"],["45046719","292292946","833645340","1372173000","1438388160","999132288","463448832","140997120","26822400","2900480","146432","2048"],["251595969","1779856128","5596799472","10298861760","12283341840","9932617728","5531344896","2110040064","537166080","86507520","7974912","344064","4096"],["1409933619","10799942322","37119747600","75518730144","101004137520","93247453728","60686106624","27917260800","8958014208","1942786560","268849152","21405696","798720","8192"],["7923848253","65336473104","243678752772","543143729856","805220624208","835990611456","623128554048","336199378944","130628289792","35879940096","6725182464","809582592","56283136","1835008","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","33","245","1921","15525","127905","1067925","9004545","76499525","653808673","5614995765","48416454529","418895174885","3634723102113"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","2"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1"]]},"terms":[{"coeff":{"coeffs":[["-2","-8","-8"],["1","4","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["9","6"],["-6","-4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166659,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-7f59bcf02d573532"]},{"id":166660,"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-7f59bcf02d573532"]},{"id":166661,"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-7f59bcf02d573532"]},{"id":166662,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-7f59bcf02d573532"]}],"region_exact":{},"incoming_edges":[{"id":"e-7f59bcf02d573532","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-60c7f085d1907563","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"e7ffddf2df79febc51e2459f661b03694a7777649ee38ab4ec0e8fea10b943cb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-78ba05b4a3dd925d","role":"sequence","name":"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.","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-e6aa9ea9917a4954","kind":"fingerprint_related","node_a_id":"n-695f6bafb476fd66","node_b_id":"n-60c7f085d1907563","fingerprint":"f6-e7d8403424fc1d9b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e7d8403424fc1d9b"},"other_id":"n-695f6bafb476fd66","other":{"id":"n-695f6bafb476fd66","name":"basis_map_A007318 of Delannoy paths counted by number of weak peaks.","first_index":0,"fingerprint":"f6-e7d8403424fc1d9b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["13","16","4"],["63","114","60","8"],["321","768","624","192","16"]],"card_rows":[["13","16","4"],["63","114","60","8"],["321","768","624","192","16"],["1683","5010","5520","2720","560","32"]],"row_sums_preview":["33","245","1921","15525","127905","1067925","9004545","76499525"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b31aa1a3fd93d174","steps":["basis A007318"],"edges":["e-9a0dabc7f010818b"],"text":"A133214 \u2192 basis A007318"},"headline":"A133214 \u2192 basis A007318","chain":null}},{"id":"l-ab42969255cdd401","kind":"oeis_match","node_a_id":"n-60c7f085d1907563","node_b_id":null,"fingerprint":null,"oeis_id":"A084771","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":15},"in_catalog":false,"oeis_id":"A084771","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A084771","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-60c7f085d1907563","oeis_id":"A084771","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":20,"oeis_terms_available":20,"term_count":200,"raw_size":19748}],"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:       3,       2;\n   3:      13,      16,        4;\n   4:      63,     114,       60,        8;\n   5:     321,     768,      624,      192,       16;\n   6:    1683,    5010,     5520,     2720,      560,      32;\n   7:    8989,   32016,    44580,    30400,    10320,    1536,      64;\n   8:   48639,  201698,   339612,   296520,   141680,   35616,    4032,    128;\n   9:  265729, 1257472,  2484160,  2646784,  1637440,  587776,  114688,  10240,   256;\n  10: 1462563, 7777314, 17635392, 22194816, 16890048, 7926912, 2236416, 350208, 25344, 512;\n","recurrence_sentence":"The row polynomials satisfy -(n-1)*P(n,x) = (n-2+(4*n-8)*x+(4*n-8)*x^2)*P(n-2,x) - (6*n-9+(4*n-6)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (4*n - 8)*x + (4*n - 8)*x^2)*P[n - 2, x] - (6*n - 9 + (4*n - 6)*x)*P[n - 1, x])/(-n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = ((n - 2 + (4*n - 8)*x + (4*n - 8)*x^2)*v[i - 2] - (6*n - 9 + (4*n - 6)*x)*v[i - 1])/(-n + 1)); 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-78ba05b4a3dd925d","steps":["basis A013609"],"edges":["e-7f59bcf02d573532"],"text":"A104684 \u2192 basis A013609"},"headline":"A104684 \u2192 basis A013609","chain":null}