{"id":"n-da7295b3362987cc","revision":0,"first_index":0,"fingerprint":"f6-487455b1f16acaff","name":"basis_map_A013611 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","20","32"],["13","176","880","1920","1536"],["63","1284","10800","47936","118272","153600","81920"],["321","8736","103296","692736","2880256","7598080","12410880","11468800","4587520"],["1683","57300","873120","7838720","45902080","183127040","503889920","943882240","1151467520","825753600","264241152"],["8989","367440","6853296","77106560","582708480","3115376640","12079575040","34217164800","70257868800","101953044480","99222552576","58133053440","15502147584"],["48639","2320388","51200240","692489280","6412688128","43003663360","215326863360","817675059200","2365862051840","5189531074560","8493283147776","10055080476672","8138627481600","4030558371840","921270484992"],["265729","14491712","369239680","5833899008","63966171136","516036362240","3168077086720","15096287068160","56420290723840","165911388487680","382549754904576","684257043283968","930632674836480","930252872220672","644428704251904","276381145497600","55276229099520"],["1462563","89749908","2593181088","46871121408","593832480768","5599657082880","40725135360000","233550426931200","1070531825172480","3951018444062720","11767737904791552","28214616718835712","54046772598669312","81583741171924992","94900993929510912","82075987507937280","49693329960468480","18793917893836800","3341140958904320"],["8097453","552196080","17840070000","363045352320","5218722485760","56324446519296","473537411481600","3175423990824960","17247944018165760","76627947568496640","279951832939233280","842459963509964800","2084427567149875200","4216862295667507200","6906482686898995200","9016078090308157440","9160800470795550720","6981314033630576640","3753771867329003520","1269633564383641600","203141370301382656"],["45046719","3379525380","120722039856","2731385152320","43922197102080","533933409343488","5095078986792960","39117999279439872","245691846885703680","1276641937046896640","5527843444322467840","20026636377364889600","60781752546502901760","154327370946733670400","326436609611897241600","571000079354418954240","816617492896064471040","938987990451172147200","846837234591491686400","576794528299488378880","278913101423798386688","85319375526580715520","12410090985684467712"],["251595969","20593500768","805941469440","20068702748160","356952200064768","4826100242608128","51534383123030016","445778267898839040","3178707329843527680","18914815797840838656","94720610026012016640","401405418368642580480","1444029144307754598400","4414142589643102617600","11453014344129293844480","25141306317853803151360","46429521963099787100160","71518375153206558720000","90760730963124619837440","93238841847780118364160","75589005459829636988928","46538875370565560958976","20445624898915160555520","5708641853414855147520","761152247121980686336"],["1409933619","125033211732","5320145344800","144583341147648","2818312429460736","41949263280258048","495604538353926144","4769260235333468160","38064446532860706816","255243899051665784832","1451411917911978147840","7044727397679271772160","29312930689547501568000","104820577707057301422080","322374455589407251496960","851969111301645863485440","1929845194991119955394560","3730183770914149483151360","6111751116675615440240640","8409190395360312589025280","9590451470992428842876928","8902880601634986528866304","6554782540303798296903680","3681883707390801074978816","1481963425146496396296192","380576123560990343168000","46840138284429580697600"],["7923848253","756798058128","34785471054768","1024511665527168","21718554076547328","352904458730643456","4570391732401238016","48430155076380819456","427744266467181920256","3191499839061372174336","20316005196088596234240","111136613435548568125440","525164260600310954721280","2151074187609152757432320","7652392635893528217518080","23658613645717387702435840","63522271707161525863055360","147812485859186089708748800","297033013152136739652894720","512774411435913553190584320","754913187056372174014119936","938404120257691261553606656","971701045163363544854429696","822715953461762508703399936","554766576467102745216679936","286497705816713530336870400","106397374113081792554598400","25293674673591973576704000","2890705676981939837337600"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","55","4525","413875","39750625","3927034375","395147828125","40277377421875","4144960687890625","429720307521484375","44812510520517578125","4695547234028857421875","493961984221113525390625","52137620317986341552734375","5518844894809846807861328125"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","20","32"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","8","16"],["-1","-8","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","-20","-32"],["6","40","64"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":205325,"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-a03f46702edbd12c"]},{"id":205326,"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-a03f46702edbd12c"]},{"id":205327,"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-a03f46702edbd12c"]}],"region_exact":{},"incoming_edges":[{"id":"e-a03f46702edbd12c","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-da7295b3362987cc","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":"0331643f6fae8dc2876ab979588bff6eb804feb35c0df109c349ee562500a11a","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,       20,         32;\n  2:      13,      176,        880,        1920,         1536;\n  3:      63,     1284,      10800,       47936,       118272,        153600,          81920;\n  4:     321,     8736,     103296,      692736,      2880256,       7598080,       12410880,        11468800,          4587520;\n  5:    1683,    57300,     873120,     7838720,     45902080,     183127040,      503889920,       943882240,       1151467520,        825753600,         264241152;\n  6:    8989,   367440,    6853296,    77106560,    582708480,    3115376640,    12079575040,     34217164800,      70257868800,     101953044480,       99222552576,       58133053440,       15502147584;\n  7:   48639,  2320388,   51200240,   692489280,   6412688128,   43003663360,   215326863360,    817675059200,    2365862051840,    5189531074560,     8493283147776,    10055080476672,     8138627481600,     4030558371840,      921270484992;\n  8:  265729, 14491712,  369239680,  5833899008,  63966171136,  516036362240,  3168077086720,  15096287068160,   56420290723840,  165911388487680,   382549754904576,   684257043283968,   930632674836480,   930252872220672,   644428704251904,   276381145497600,    55276229099520;\n  9: 1462563, 89749908, 2593181088, 46871121408, 593832480768, 5599657082880, 40725135360000, 233550426931200, 1070531825172480, 3951018444062720, 11767737904791552, 28214616718835712, 54046772598669312, 81583741171924992, 94900993929510912, 82075987507937280, 49693329960468480, 18793917893836800, 3341140958904320;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n-1+(8*n-8)*x+(16*n-16)*x^2)*P(n-2,x) + (6*n-3+(40*n-20)*x+(64*n-32)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 20*x + 32*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 20*x + 32*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (8*n - 8)*x + (16*n - 16)*x^2)*P[n - 2, x] + (6*n - 3 + (40*n - 20)*x + (64*n - 32)*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 + 20*x + 32*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (8*n - 8)*x + (16*n - 16)*x^2)*v[i - 2] + (6*n - 3 + (40*n - 20)*x + (64*n - 32)*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 A013611"],"edges":["e-a03f46702edbd12c"],"text":"A109983 \u2192 basis A013611"},"headline":"A109983 \u2192 basis A013611","chain":null}