{"id":"n-d9640fb05f96c50b","revision":0,"first_index":0,"fingerprint":"f6-a7f447fa676c998a","name":"shift_x_plus_one of Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","5","2"],["13","44","55","30","6"],["63","321","675","749","462","150","20"],["321","2184","6456","10824","11251","7420","3030","700","70"],["1683","14325","54570","122480","179305","178835","123020","57610","17570","3150","252"],["8989","91860","428331","1204790","2276205","3042360","2949115","2088450","1072050","388920","94626","13860","924"],["48639","580097","3200015","10820145","25049563","41995765","52570035","49906925","36100190","19796490","8099826","2397318","485100","60060","3432"],["265729","3622928","23077480","91154672","249867856","503941760","773456320","921404240","860905315","632901720","364827876","163139592","55470030","13861848","2400684","257400","12870"],["1462563","22437477","162073818","732361272","2319658128","5468415120","9942660000","14254786800","16335019305","15071939255","11222589402","6726888828","3221438682","1215692478","353533752","76439220","11570130","1093950","48620"],["8097453","138049020","1115004375","5672583630","20385634710","55004342304","115609719600","193812499440","263182739535","292312422060","266982872905","200858107450","124241564700","62836144800","25728652950","8396877060","2132915070","406365960","54624570","4618900","184756"],["45046719","844881345","7545127491","42677893005","171571082430","521419345062","1243915768260","2387573198208","3748960066005","4870002506435","5271762317965","4774722189275","3622874769360","2299657031100","1216071134850","531785264010","190133576490","54656294550","12323103650","2098366270","253669988","19399380","705432"],["251595969","5148375192","50371341840","313573480440","1394344531503","4712988518172","12581636504646","27208146234060","48503224637505","72154296103824","90332613016140","95702509491120","86070844191775","65775850260900","42665803224330","23414666129140","10810215483210","4162917330000","1320742463040","339200930640","68747799978","10581715144","1162199220","81124680","2704156"],["1409933619","31258302933","332509084050","2259114705432","11009032927581","40966077422127","120997201746564","291092543660490","580817360425731","973678203779853","1384174268638590","1679593896312540","1747186821076125","1561948324845095","1200938431879160","793458066230310","449327098902110","217125272082290","88937684146740","30592456443120","8722464800478","2024280684426","372596249480","52322714444","5264991732","338019500","10400600"],["7923848253","189199514532","2174091940923","16007994773862","84838101861513","344633260479144","1115818294043271","2955942082298634","6526859534716521","12174605709310044","19374852367485615","26497033461463110","31302229201812205","32053503209488880","28507384046515555","22033800972362410","14789931407934410","8603819055668200","4322399227416270","1865462441622780","686589543926286","213368393874064","55234809517706","11691496886524","1970926804756","254461079600","23624962900","1404081000","40116600"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","10","148","2440","42256","752800","13660480","251113600","4660568320","87140108800","1638884021248","30970912737280","587599919386624","11185644310405120","213540626285805568"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","6","2"],["-1","-1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","10","8","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-fbebeadd27b9d988","recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","5","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","2","1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","-5","-2"],["6","10","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178256,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-94901725fd2b2473"]},{"id":178257,"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-16a835f21d117a04","e-94901725fd2b2473"]},{"id":178258,"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-16a835f21d117a04","e-94901725fd2b2473"]},{"id":178259,"position":3,"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-16a835f21d117a04"]}],"region_exact":{},"incoming_edges":[{"id":"e-16a835f21d117a04","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-d9640fb05f96c50b","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":"06112203c6494ab76efac8d454e12766959aab2ff2bc2c84161e33461c7910f8","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-e4d248920a317259","role":"sequence","name":"Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","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]"}]},{"id":"e-fbebeadd27b9d988","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-d9640fb05f96c50b","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"dfb1ae0d514e2ca396bab9d26754ddad31028a56c959cb5d3f9b54b80d0b1231","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-94901725fd2b2473","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-d9640fb05f96c50b","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"d2a199f22f13b3f32242f9498ee8e3f2cc5fe52190a76eb4b86ba8f13a93ad49","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-e4d248920a317259","role":"sequence","name":"Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","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:       1;\n  1:       3,        5,         2;\n  2:      13,       44,        55,        30,          6;\n  3:      63,      321,       675,       749,        462,        150,         20;\n  4:     321,     2184,      6456,     10824,      11251,       7420,       3030,         700,          70;\n  5:    1683,    14325,     54570,    122480,     179305,     178835,     123020,       57610,       17570,        3150,         252;\n  6:    8989,    91860,    428331,   1204790,    2276205,    3042360,    2949115,     2088450,     1072050,      388920,       94626,      13860,        924;\n  7:   48639,   580097,   3200015,  10820145,   25049563,   41995765,   52570035,    49906925,    36100190,    19796490,     8099826,    2397318,     485100,      60060,      3432;\n  8:  265729,  3622928,  23077480,  91154672,  249867856,  503941760,  773456320,   921404240,   860905315,   632901720,   364827876,  163139592,   55470030,   13861848,   2400684,   257400,    12870;\n  9: 1462563, 22437477, 162073818, 732361272, 2319658128, 5468415120, 9942660000, 14254786800, 16335019305, 15071939255, 11222589402, 6726888828, 3221438682, 1215692478, 353533752, 76439220, 11570130, 1093950, 48620;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n-4+(n-6)*x-2*x^2)*P(n-1,x) + (4+10*x+8*x^2+2*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 4 + (n - 6)*x - 2*x^2)*P[n - 1, x] + (4 + 10*x + 8*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(n - 4 + (n - 6)*x - 2*x^2)*v[i - 1] + (4 + 10*x + 8*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n)); 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-e4d248920a317259","steps":["basis A007318"],"edges":["e-16a835f21d117a04"],"text":"A109983 \u2192 basis A007318"},"headline":"A109983 \u2192 basis A007318","chain":null}