{"id":"n-9906787568ecda50","revision":0,"first_index":0,"fingerprint":"f6-c986abc9710a95e7","name":"basis_map_A013610 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","15","18"],["13","132","495","810","486"],["63","963","6075","20223","37422","36450","14580"],["321","6552","58104","292248","911331","1803060","2208870","1530900","459270"],["1683","42975","491130","3306960","14523705","43456905","89681580","125993070","115276770","62001450","14880348"],["8989","275580","3854979","32529330","184372605","739293480","2149904835","4567440150","7033720050","7655112360","5587570674","2455257420","491051484"],["48639","1740291","28800135","292143915","2029014603","10204970895","38323555515","109146444975","236853346590","389654312670","478286625474","424677691746","257802029100","95755039380","16415149608"],["265729","10868784","207697320","2461176144","20239296336","122457847680","563849657280","2015111072880","5648399771715","12457404554760","21542721249924","28899689304024","29479048213230","22100263088904","11482397150796","3693408661800","554011299270"],["1462563","67312431","1458664362","19773754344","187892308368","1328824874160","7248199140000","31175218731600","107174061660105","296660980356165","662682681598698","1191648175213716","1712004594600762","1938206478602394","1690940976269688","1096819258932540","498056158043730","141272881313850","18836384175180"],["8097453","414147060","10035039375","153159758010","1651236411510","13366055179872","84279485588400","423867936275280","1726741954089135","5753585403406980","15765071662167345","35581411160445150","66027061385732700","100181110885970400","123059349471608550","120486008024373420","91814999934985470","52478166312051480","21162677620814730","5368369489926300","644204338791156"],["45046719","2534644035","67906147419","1152303111135","13897257676830","126704900850066","906814595061540","5221622584480896","24596926993058805","95856259334160105","311292293113515285","845827711663498425","1925344190303447760","3666396096794445300","5816430539782369650","7630537297249937070","8184627019897189290","7058322787163011650","4774222842080684850","2438850259273518090","884492557160257188","202924366719214140","22137203642096088"],["251595969","15445125576","453342076560","8466483971880","112941907051743","1145256209915796","9172013011886934","59504215813889220","318229656846670305","1420213010211567792","5334050465990050860","16953412448823434640","45741575508121097775","104867950915508870700","204069214182070435770","335974866723079849980","465344329855621054410","537599822551724790000","511682690874021226560","394240171253411648880","239708756566358543178","110688477899774006232","36471043000353304980","7637335256523150360","763733525652315036"],["1409933619","93774908799","2992581756450","60996097046664","891731667134061","9954756813576861","88206960073245156","636619392985491630","3810742701753221091","19164908084998846599","81734106388840100910","297535019950077523380","928526711379516946125","2490250139112006395685","5744051290586634026040","11385256000738558771170","19342058264178535481310","28039593028126280013270","34456281082657558555860","35556433304714253257040","30413354204578267743678","21174690941106541316478","11692465115021715253320","4925826662615547877188","1486989174445057375092","286400072119618138500","26436929734118597400"],["7923848253","567598543596","19566827468307","432215858894274","6871886250782553","83745882296431992","813431536357544559","6464645333987112558","42822725407275094281","239632764176349596052","1144065657447658080135","4693869986597805547170","16635287989240280037405","51103637397461939628240","136349934165578457582795","316160961008937791385870","636658050926489733569610","1111098595271497421916600","1674586022338833529956030","2168155114033894143418260","2393989711651878318664686","2231908762279530941026992","1733326849966892526936954","1100674682143342658427348","556647943905215466303636","215601974291648534662800","60051485891050393994100","10706956542318031947000","917739132198688452600"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","36","1936","115776","7270656","469656576","30900391936","2059470913536","138581681176576","9394264067014656","640572085974859776","43888128613297422336","3018885817056557531136","208351593435165282533376","14420701469216635046854656"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","15","18"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","6","9"],["-1","-6","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","-15","-18"],["6","30","36"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":202500,"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-5e8dce24aa3bcfcc"]},{"id":202501,"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-5e8dce24aa3bcfcc"]},{"id":202502,"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-5e8dce24aa3bcfcc"]}],"region_exact":{},"incoming_edges":[{"id":"e-5e8dce24aa3bcfcc","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-9906787568ecda50","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":"e4a9d4b41bd1ff791cef8a7f66c024ee559b6109551f637ecfb7ac3a53f4b1c2","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]"}]}],"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,       15,         18;\n  2:      13,      132,        495,         810,          486;\n  3:      63,      963,       6075,       20223,        37422,         36450,         14580;\n  4:     321,     6552,      58104,      292248,       911331,       1803060,       2208870,        1530900,          459270;\n  5:    1683,    42975,     491130,     3306960,     14523705,      43456905,      89681580,      125993070,       115276770,        62001450,        14880348;\n  6:    8989,   275580,    3854979,    32529330,    184372605,     739293480,    2149904835,     4567440150,      7033720050,      7655112360,      5587570674,       2455257420,        491051484;\n  7:   48639,  1740291,   28800135,   292143915,   2029014603,   10204970895,   38323555515,   109146444975,    236853346590,    389654312670,    478286625474,     424677691746,     257802029100,      95755039380,      16415149608;\n  8:  265729, 10868784,  207697320,  2461176144,  20239296336,  122457847680,  563849657280,  2015111072880,   5648399771715,  12457404554760,  21542721249924,   28899689304024,   29479048213230,   22100263088904,   11482397150796,    3693408661800,    554011299270;\n  9: 1462563, 67312431, 1458664362, 19773754344, 187892308368, 1328824874160, 7248199140000, 31175218731600, 107174061660105, 296660980356165, 662682681598698, 1191648175213716, 1712004594600762, 1938206478602394, 1690940976269688, 1096819258932540, 498056158043730, 141272881313850, 18836384175180;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n-1+(6*n-6)*x+(9*n-9)*x^2)*P(n-2,x) + (6*n-3+(30*n-15)*x+(36*n-18)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 15*x + 18*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 15*x + 18*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (6*n - 6)*x + (9*n - 9)*x^2)*P[n - 2, x] + (6*n - 3 + (30*n - 15)*x + (36*n - 18)*x^2)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 15*x + 18*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (6*n - 6)*x + (9*n - 9)*x^2)*v[i - 2] + (6*n - 3 + (30*n - 15)*x + (36*n - 18)*x^2)*v[i - 1])/(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 A013610"],"edges":["e-5e8dce24aa3bcfcc"],"text":"A109983 \u2192 basis A013610"},"headline":"A109983 \u2192 basis A013610","chain":null}