{"id":"n-5b112963159320a9","revision":1,"first_index":1,"fingerprint":"f6-96e61be3a8e09486","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","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","3"],["11","24","9"],["45","153","135","27"],["197","912","1314","648","81"],["903","5295","10710","8910","2835","243"],["4279","30384","79515","95040","52245","11664","729"],["20793","173397","558117","877905","705915","277263","45927","2187"],["103049","987072","3775716","7399728","7939134","4626720","1367604","174960","6561"],["518859","5613435","24894540","58571100","79147530","62246394","27617436","6377292","649539","19683"],["2646723","31917960","161081325","442972800","724881150","724626000","437990490","153264960","28441935","2361960","59049"],["13648869","181535409","1027547235","3237114375","6235708050","7605460170","5888913030","2829824910","802180665","122329845","8444007","177147"],["71039373","1033029072","6482418822","23032029240","51126938775","73864612800","70237298340","43527773520","17060141835","4001160240","510537654","29760696","531441"],["372693519","5882255847","40534478946","160413499818","403609669749","675494185509","765048300588","585574997436","297715984137","97108895169","19167895890","2077225722","103630995","1594323"],["1968801519","33518268000","251640571455","1097988877440","3090213801675","5888434327776","7762106210859","7110919715328","4487620998861","1908480686112","526593608541","88726555008","8269753401","357128352","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","44","360","3152","28896","273856","2661504","26380544","265655808","2710244352","27952883712","290977271808","3053105307648","32256844087296"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","3"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","-12","18"],["-1","6","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","-3"],["6","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167294,"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-f8ee48cf56436bdf"]},{"id":167295,"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-f8ee48cf56436bdf"]},{"id":167296,"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-f8ee48cf56436bdf"]},{"id":167297,"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-f8ee48cf56436bdf"]}],"region_exact":{},"incoming_edges":[{"id":"e-f8ee48cf56436bdf","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-5b112963159320a9","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":"1684de78f99b73c72fb3eb51efc594fe6ec233d38dcaaf70232a96fa99998fc1","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-0d75f6272d2a2a84","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","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-4024100774329161","kind":"oeis_match","node_a_id":"n-5b112963159320a9","node_b_id":null,"fingerprint":null,"oeis_id":"A090442","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":15},"in_catalog":false,"oeis_id":"A090442","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A090442","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-5b112963159320a9","oeis_id":"A090442","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":21726}],"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,       3;\n   3:     11,      24,        9;\n   4:     45,     153,      135,       27;\n   5:    197,     912,     1314,      648,       81;\n   6:    903,    5295,    10710,     8910,     2835,      243;\n   7:   4279,   30384,    79515,    95040,    52245,    11664,      729;\n   8:  20793,  173397,   558117,   877905,   705915,   277263,    45927,    2187;\n   9: 103049,  987072,  3775716,  7399728,  7939134,  4626720,  1367604,  174960,   6561;\n  10: 518859, 5613435, 24894540, 58571100, 79147530, 62246394, 27617436, 6377292, 649539, 19683;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = -(n-2-(6*n-12)*x+(9*n-18)*x^2)*P(n-2,x) + (6*n-3+(6*n-3)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 - (6*n - 12)*x + (9*n - 18)*x^2)*P[n - 2, x] + (6*n - 3 + (6*n - 3)*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 + 3*x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 - (6*n - 12)*x + (9*n - 18)*x^2)*v[i - 2] + (6*n - 3 + (6*n - 3)*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-0d75f6272d2a2a84","steps":["basis A013610"],"edges":["e-f8ee48cf56436bdf"],"text":"A114656 \u2192 basis A013610"},"headline":"A114656 \u2192 basis A013610","chain":null}
