{"id":"n-b31aa1a3fd93d174","revision":1,"first_index":0,"fingerprint":"f6-806d331b1bf3c1f4","name":"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":[["1","8","4"],["1","18","36","8"],["1","32","144","128","16"],["1","50","400","800","400","32"],["1","72","900","3200","3600","1152","64"],["1","98","1764","9800","19600","14112","3136","128"],["1","128","3136","25088","78400","100352","50176","8192","256"],["1","162","5184","56448","254016","508032","451584","165888","20736","512"],["1","200","8100","115200","705600","2032128","2822400","1843200","518400","51200","1024"],["1","242","12100","217800","1742400","6830208","13660416","13939200","6969600","1548800","123904","2048"],["1","288","17424","387200","3920400","20072448","54641664","80289792","62726400","24780800","4460544","294912","4096"],["1","338","24336","654368","8179600","53003808","188457984","376915968","424030464","261747200","83759104","12460032","692224","8192"],["1","392","33124","1059968","16032016","128256128","577152576","1507663872","2308610304","2052098048","1026049024","271351808","33918976","1605632","16384"],["1","450","44100","1656200","29811600","288576288","1603201600","5300380800","10600761600","12825612800","9234441216","3815884800","847974400","90316800","3686400","32768"],["1","512","57600","2508800","52998400","610541568","4104196096","16751820800","42403046400","67007283200","65667137536","39074660352","13567590400","2569011200","235929600","8388608","65536"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["13","63","321","1683","8989","48639","265729","1462563","8097453","45046719","251595969","1409933619","7923848253","44642381823","252055236609"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A133214","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","8","4"],["1","18","36","8"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["3","6"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","4","-4"],["-1","4","-4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159242,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-b87132cd573fb9e2"]},{"id":159243,"position":1,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"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-4ab8ec5aa6b36e54"]},{"id":159244,"position":2,"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-4ab8ec5aa6b36e54"]},{"id":159245,"position":3,"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-4ab8ec5aa6b36e54"]},{"id":159246,"position":4,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-4ab8ec5aa6b36e54"]}],"region_exact":{},"incoming_edges":[{"id":"e-4ab8ec5aa6b36e54","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-b31aa1a3fd93d174","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"263aad9fe9144f6ffc197c3d1e26ddde45dafc1425ebe066de684c6b4ebdf452","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},{"id":"e-b87132cd573fb9e2","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-b31aa1a3fd93d174","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"83e7ad4e4ce3b9b5b1ca32b94dd205c6f0771a0bc7433064eb03fa37d37ce223","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[]}],"used_by":[{"id":"e-033100947c42f87a","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-bf4beaa81304b08d","conclusion_name":"basis_map_A008277 of Delannoy paths counted by number of weak peaks."},{"id":"e-3165932d1e38479b","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-e1b5185a7301abc6","conclusion_name":"basis_map_A013609 of Delannoy paths counted by number of weak peaks."},{"id":"e-8c219f5be19a6675","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-4acec5e35bbb846a","conclusion_name":"basis_map_A013611 of Delannoy paths counted by number of weak peaks."},{"id":"e-9a0dabc7f010818b","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-695f6bafb476fd66","conclusion_name":"basis_map_A007318 of Delannoy paths counted by number of weak peaks."},{"id":"e-f0f0fb352cf0ac03","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-0180c64ef6975bff","conclusion_name":"basis_map_A013610 of Delannoy paths counted by number of weak peaks."}],"used_by_total":5,"links":[{"id":"l-543a07a4597cfa92","kind":"oeis_match","node_a_id":"n-b31aa1a3fd93d174","node_b_id":null,"fingerprint":null,"oeis_id":"A001850","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":"A001850","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001850","relation":"match_row_sums"},{"oeis_id":"A133214","relation":"tail"}],"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: 1,   8,     4;\n  1: 1,  18,    36,      8;\n  2: 1,  32,   144,    128,      16;\n  3: 1,  50,   400,    800,     400,      32;\n  4: 1,  72,   900,   3200,    3600,    1152,       64;\n  5: 1,  98,  1764,   9800,   19600,   14112,     3136,      128;\n  6: 1, 128,  3136,  25088,   78400,  100352,    50176,     8192,     256;\n  7: 1, 162,  5184,  56448,  254016,  508032,   451584,   165888,   20736,     512;\n  8: 1, 200,  8100, 115200,  705600, 2032128,  2822400,  1843200,  518400,   51200,   1024;\n  9: 1, 242, 12100, 217800, 1742400, 6830208, 13660416, 13939200, 6969600, 1548800, 123904, 2048;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (2*n+3+(4*n+6)*x)*P(n-1,x) - (n+1-(4*n+4)*x+(4*n+4)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1 + 8*x + 4*x^2, P(1,x) = 1 + 18*x + 36*x^2 + 8*x^3.","mathematica":"P[0, x_] := 1 + 8*x + 4*x^2;\nP[1, x_] := 1 + 18*x + 36*x^2 + 8*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 3 + (4*n + 6)*x)*P[n - 1, x] - (n + 1 - (4*n + 4)*x + (4*n + 4)*x^2)*P[n - 2, x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 8*x + 4*x^2; v[2] = 1 + 18*x + 36*x^2 + 8*x^3; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 3 + (4*n + 6)*x)*v[i - 1] - (n + 1 - (4*n + 4)*x + (4*n + 4)*x^2)*v[i - 2])/(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":[],"edges":[],"text":"A133214"},"headline":"Delannoy paths counted by number of weak peaks.","chain":null}