{"id":"n-fa10d6b5292a8c39","revision":0,"first_index":0,"fingerprint":"f6-e7d8403424fc1d9b","name":"reversal of basis_map_A007318 of Delannoy paths counted by number of weak peaks.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["4","16","13"],["8","60","114","63"],["16","192","624","768","321"],["32","560","2720","5520","5010","1683"],["64","1536","10320","30400","44580","32016","8989"],["128","4032","35616","141680","296520","339612","201698","48639"],["256","10240","114688","587776","1637440","2646784","2484160","1257472","265729"],["512","25344","350208","2236416","7926912","16890048","22194816","17635392","7777314","1462563"],["1024","61440","1025280","7956480","34755840","93413376","160668480","177672960","122356260","47800080","8097453"],["2048","146432","2900480","26822400","140997120","463448832","999132288","1438388160","1372173000","833645340","292292946","45046719"],["4096","344064","7974912","86507520","537166080","2110040064","5531344896","9932617728","12283341840","10298861760","5596799472","1779856128","251595969"],["8192","798720","21405696","268849152","1942786560","8958014208","27917260800","60686106624","93247453728","101004137520","75518730144","37119747600","10799942322","1409933619"],["16384","1835008","56283136","809582592","6725182464","35879940096","130628289792","336199378944","623128554048","835990611456","805220624208","543143729856","243678752772","65336473104","7923848253"],["32768","4177920","145367040","2372464640","22422865920","136794577920","573718745600","1718468294400","3758645404800","6064796337600","7216215630624","6255647863920","3843073820040","1585742494140","394246725570","44642381823"],["65536","9437184","369623040","6789529600","72370421760","499883704320","2387567902720","8209563648000","20833404134400","39561328435200","56475739127808","60344781299712","47547385616640","26814614400000","10241659272960","2373580947456","252055236609"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["33","245","1921","15525","127905","1067925","9004545","76499525","653808673","5614995765","48416454529","418895174885","3634723102113","31616937184725","275621102802945"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4","16","13"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","4","-2"],["2","1","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187648,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-7234a951849608e0"]},{"id":187649,"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-7234a951849608e0"]},{"id":187650,"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-7234a951849608e0"]}],"region_exact":{},"incoming_edges":[{"id":"e-7234a951849608e0","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-fa10d6b5292a8c39","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"423d85d1dfcf949588b1df605c0750b789e149eb2b7a19857d104e75ba6c8f65","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-695f6bafb476fd66","role":"sequence","name":"basis_map_A007318 of Delannoy paths counted by number of weak peaks.","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-274e0c72bf82b81b","kind":"fingerprint_related","node_a_id":"n-fa10d6b5292a8c39","node_b_id":"n-9a53de4e8b612343","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-9a53de4e8b612343","other":{"id":"n-9a53de4e8b612343","name":"basis_map_A013609 of basis_map_A007318 of Square the entries of Pascal's triangle.","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":[["1"],["3","2"],["13","16","4"]],"card_rows":[["1"],["3","2"],["13","16","4"],["63","114","60","8"]],"row_sums_preview":["1","5","33","245","1921","15525","127905","1067925"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b13531ad79e7c629","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-041c07f9bf4043d7","e-8cc81e7818a84f55"],"text":"A008459 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A008459 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}},{"id":"l-45713dbe3c552e48","kind":"fingerprint_related","node_a_id":"n-fa10d6b5292a8c39","node_b_id":"n-6ebbdbbb1ecce33b","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-6ebbdbbb1ecce33b","other":{"id":"n-6ebbdbbb1ecce33b","name":"basis_map_A013609 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-e7d8403424fc1d9b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["78","96","24"]],"card_rows":[["1"],["6","4"],["78","96","24"],["1260","2280","1200","160"]],"row_sums_preview":["1","10","198","4900","134470","3912300","118184220","3665118600"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fdc03fad975441d0","steps":["basis A007318","basis A013609"],"edges":["e-2cd9cd27e716475b","e-ba8e5a81d3af18d8"],"text":"A069466 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A069466 \u2192 basis A007318 \u2192 basis A013609","chain":null}},{"id":"l-637eecf4320f2051","kind":"fingerprint_related","node_a_id":"n-83a30b01f0214b4b","node_b_id":"n-fa10d6b5292a8c39","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-83a30b01f0214b4b","other":{"id":"n-83a30b01f0214b4b","name":"reversal of 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.","first_index":1,"fingerprint":"f6-e7d8403424fc1d9b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["4","16","13"]],"card_rows":[["1"],["2","3"],["4","16","13"],["8","60","114","63"]],"row_sums_preview":["1","5","33","245","1921","15525","127905","1067925"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-78ba05b4a3dd925d","steps":["basis A013609","reverse"],"edges":["e-2f2f885dba2c5644","e-066da29b71efe1d7"],"text":"A104684 \u2192 basis A013609 \u2192 reverse"},"headline":"A104684 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-b782b2b2b601a796","kind":"fingerprint_related","node_a_id":"n-695f6bafb476fd66","node_b_id":"n-fa10d6b5292a8c39","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]","(-inf, -1]"],"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-c755879d3c3c0742","kind":"fingerprint_related","node_a_id":"n-60c7f085d1907563","node_b_id":"n-fa10d6b5292a8c39","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-60c7f085d1907563","other":{"id":"n-60c7f085d1907563","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.","first_index":1,"fingerprint":"f6-e7d8403424fc1d9b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["13","16","4"]],"card_rows":[["1"],["3","2"],["13","16","4"],["63","114","60","8"]],"row_sums_preview":["1","5","33","245","1921","15525","127905","1067925"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-78ba05b4a3dd925d","steps":["basis A013609"],"edges":["e-2f2f885dba2c5644"],"text":"A104684 \u2192 basis A013609"},"headline":"A104684 \u2192 basis A013609","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:    4,     16,      13;\n  1:    8,     60,     114,       63;\n  2:   16,    192,     624,      768,       321;\n  3:   32,    560,    2720,     5520,      5010,      1683;\n  4:   64,   1536,   10320,    30400,     44580,     32016,      8989;\n  5:  128,   4032,   35616,   141680,    296520,    339612,    201698,      48639;\n  6:  256,  10240,  114688,   587776,   1637440,   2646784,   2484160,    1257472,     265729;\n  7:  512,  25344,  350208,  2236416,   7926912,  16890048,  22194816,   17635392,    7777314,   1462563;\n  8: 1024,  61440, 1025280,  7956480,  34755840,  93413376, 160668480,  177672960,  122356260,  47800080,   8097453;\n  9: 2048, 146432, 2900480, 26822400, 140997120, 463448832, 999132288, 1438388160, 1372173000, 833645340, 292292946, 45046719;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (2*n+4+(n+4)*x-(2*n+2)*x^2)*P(n-1,x) + (4*x+6*x^2+2*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 16*x + 13*x^2.","mathematica":"P[0, x_] := 4 + 16*x + 13*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 4 + (n + 4)*x - (2*n + 2)*x^2)*P[n - 1, x] + (4*x + 6*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4 + 16*x + 13*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 4 + (n + 4)*x - (2*n + 2)*x^2)*v[i - 1] + (4*x + 6*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 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}}],"derivation":{"root":"n-b31aa1a3fd93d174","steps":["basis A007318","reverse"],"edges":["e-9a0dabc7f010818b","e-7234a951849608e0"],"text":"A133214 \u2192 basis A007318 \u2192 reverse"},"headline":"A133214 \u2192 basis A007318 \u2192 reverse","chain":null}