{"id":"n-fb83cf6de9ce04de","revision":1,"first_index":0,"fingerprint":"f6-dae6101d606a19f7","name":"basis_map_A013610 of Triangle of Dyck paths counted by number of long interior inclines.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2"],["5","6"],["14","36","18"],["42","168","189","54"],["132","720","1296","864","162"],["429","2970","7425","8100","3645","486"],["1430","12012","38610","59400","44550","14580","1458"],["4862","48048","189189","378378","405405","224532","56133","4374"],["16796","190944","891072","2201472","3095820","2476656","1061424","209952","13122"],["58786","755820","4081428","12029472","21051576","22289904","13931190","4776408","767637","39366"],["208012","2984520","18314100","62791200","131861520","174890016","145741680","73483200","20667150","2755620","118098"],["742900","11767536","80901810","316572300","777041100","1243265760","1305429048","883373040","368072100","86605200","9743085","354294"],["2674440","46356960","353026080","1553314752","4368697740","8205554016","10443432384","8951513472","5035226328","1766746080","353349216","34012224","1062882"],["9694845","182530530","1525433715","7457675940","23663779425","51113763558","76670645337","80004151656","57275699481","27274142610","8182242783","1409381532","117448461","3188646"],["35357670","718559100","6538887810","35174706840","124367713470","304009966260","526171095450","649445466384","568264783086","345900302748","141504669306","36754459560","5513168934","401769396","9565938"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","11","68","453","3174","23055","172040","1310921","10157258","79781587","633815116","5083868173","41114966574","334889778519","2744810471952"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["5","6"]],"recurrence":{"denominator":{"coeffs":[["3"],["7"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0","27"],["0","0","-9"],["0","0","-18"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","0"],["16","24"],["8","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167182,"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-7d811e606f00941d"]},{"id":167183,"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-7d811e606f00941d"]},{"id":167184,"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-7d811e606f00941d"]},{"id":167185,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-7d811e606f00941d"]}],"region_exact":{},"incoming_edges":[{"id":"e-7d811e606f00941d","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-fb83cf6de9ce04de","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":"05f0fd70691ec9faa3d0678b99a6876f3e182423e7619f4813f12702c3823fc1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c963abfc0266d66c","role":"sequence","name":"Triangle of Dyck paths counted by number of long interior inclines.","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":[],"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:     2;\n  1:     5,      6;\n  2:    14,     36,      18;\n  3:    42,    168,     189,       54;\n  4:   132,    720,    1296,      864,      162;\n  5:   429,   2970,    7425,     8100,     3645,      486;\n  6:  1430,  12012,   38610,    59400,    44550,    14580,     1458;\n  7:  4862,  48048,  189189,   378378,   405405,   224532,    56133,    4374;\n  8: 16796, 190944,  891072,  2201472,  3095820,  2476656,  1061424,  209952,  13122;\n  9: 58786, 755820, 4081428, 12029472, 21051576, 22289904, 13931190, 4776408, 767637, 39366;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+7*n+3)*P(n,x) = -(18*n^2+9*n-27)*x^2*P(n-2,x) + (8*n^2+16*n+6+(12*n^2+24*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 5 + 6*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 5 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(18*n^2 + 9*n - 27)*x^2*P[n - 2, x] + (8*n^2 + 16*n + 6 + (12*n^2 + 24*n)*x)*P[n - 1, x])/(2*n^2 + 7*n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 5 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(18*n^2 + 9*n - 27)*x^2*v[i - 2] + (8*n^2 + 16*n + 6 + (12*n^2 + 24*n)*x)*v[i - 1])/(2*n^2 + 7*n + 3)); 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-c963abfc0266d66c","steps":["basis A013610"],"edges":["e-7d811e606f00941d"],"text":"A108838 \u2192 basis A013610"},"headline":"A108838 \u2192 basis A013610","chain":null}