{"id":"n-5fedd6b839deb652","revision":0,"first_index":0,"fingerprint":"f6-a1bc1a919c7f051b","name":"one_plus_derivative of basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","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"],["20","30","8"],["204","752","964","480","64"],["1728","10296","24720","30200","19296","5760","512"],["13392","110592","391968","774784","926736","677632","288256","61440","4096"],["98496","1041120","4890240","13408320","23682880","28013024","22293120","11627520","3706880","614400","32768"],["699840","9020160","52809408","185448960","434260800","712608000","837344320","706263552","421002240","170639360","43499520","5898240","262144"],["4852224","73809792","517757184","2218043520","6476004864","13613314176","21208748288","24815796096","21837831680","14309349376","6817800192","2259582976","479854592","55050240","2097152"],["33032448","579280896","4736517120","23955886080","83826925056","215009206272","417706791936","626123509760","730506461440","663951826944","466921570304","250110672896","99241820160","27818721280","5062524928","503316480","16777216"],["221709312","4403953152","41130436608","239889475584","978675664896","2964140282880","6902446731264","12625313605632","18368827610112","21387462516224","19939656812544","14813410590720","8674147958784","3928985174016","1335616339968","324362305536","51640270848","4529848320","134217728"],["1471343616","32651735040","342943994880","2266093117440","10561806120960","36894406238208","100178390476800","216390232719360","377389509073920","536199414231040","623380202898432","593244624609280","460489761177600","289248769474560","145083169505280","56924426993664","16925400760320","3625891921920","513047265280","40265318400","1073741824"],["9674588160","237228963840","2767819051008","20436425349120","107134866432000","424051576565760","1315568423854080","3277760647163904","6666007281500160","11185186640517120","15587932329840640","18101850168227840","17521111203864576","14097115597537280","9372454848430080","5099010502164480","2236729440337920","774027211702272","204506545520640","39190905487360","4990215127040","354334801920","8589934592"],["63126687744","1694987845632","21749925371904","177473870069760","1033646946545664","4572187575386112","15956140126519296","45049809728765952","104686506248417280","202622848065994752","329292775732174848","451608159047614464","523980711482626048","514429509645434880","426463547638284288","297169456328605696","172772352184025088","82894435569893376","32308280457953280","10000054496526336","2377294194671616","412114996953088","47699906789376","3092376453120","68719476736"],["409235079168","11941827895296","167147859640320","1493661278453760","9568894251810816","46780822689030144","181393875956367360","572380561328209920","1496300774185451520","3282004628241850368","6094771344838066176","9641985149153230848","13045828057432801280","15126012702260060160","15030949852202565632","12780143096614879232","9264411914768744448","5692878512728309760","2940561929829089280","1261795491063005184","442449132804636672","123868425016573952","26764163866951680","4235610847969280","449356658376704","26800595927040","549755813888"],["2638260191232","83135670976512","1260504993742848","12243021043728384","85562016731283456","458143219474956288","1954172247067901952","6815689718527033344","19797968424932425728","48535470783271600128","101398209542471761920","181791576590897774592","281067441169042784256","375919852063239766016","435618824446774362112","437412521458527240192","380073542370007449600","284965605928433876992","183519416774220054528","100850515772151169024","46863260473104531456","18187188499101253632","5795807414234120192","1481138842328104960","293305051810627584","42698709430960128","4181167842525184","230897441832960","4398046511104"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","58","2464","92512","3248896","109408768","3579756544","114689892352","3616104841216","112584995504128","3469733552717824","106038442476961792","3217881673379086336","97067234807290789888","2912945917279080546304"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-4","-1","4"],["7","8","0"]]},"terms":[{"coeff":{"coeffs":[["18","84","146","112","32"],["42","146","168","64","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192044,"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-9d563bdd96ef98c8"]},{"id":192045,"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-9d563bdd96ef98c8"]},{"id":192046,"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-9d563bdd96ef98c8"]}],"region_exact":{},"incoming_edges":[{"id":"e-9d563bdd96ef98c8","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-5fedd6b839deb652","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"b8a9a19ab2d607c816d7d89a32389f43b0b0c98e029c03d635efd75f1af02dc9","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-83675781d9687218","role":"sequence","name":"basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","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:        20,         30,           8;\n  2:       204,        752,         964,          480,           64;\n  3:      1728,      10296,       24720,        30200,        19296,          5760,           512;\n  4:     13392,     110592,      391968,       774784,       926736,        677632,        288256,          61440,           4096;\n  5:     98496,    1041120,     4890240,     13408320,     23682880,      28013024,      22293120,       11627520,        3706880,         614400,          32768;\n  6:    699840,    9020160,    52809408,    185448960,    434260800,     712608000,     837344320,      706263552,      421002240,      170639360,       43499520,        5898240,        262144;\n  7:   4852224,   73809792,   517757184,   2218043520,   6476004864,   13613314176,   21208748288,    24815796096,    21837831680,    14309349376,     6817800192,     2259582976,     479854592,      55050240,       2097152;\n  8:  33032448,  579280896,  4736517120,  23955886080,  83826925056,  215009206272,  417706791936,   626123509760,   730506461440,   663951826944,   466921570304,   250110672896,   99241820160,   27818721280,    5062524928,    503316480,    16777216;\n  9: 221709312, 4403953152, 41130436608, 239889475584, 978675664896, 2964140282880, 6902446731264, 12625313605632, 18368827610112, 21387462516224, 19939656812544, 14813410590720, 8674147958784, 3928985174016, 1335616339968, 324362305536, 51640270848, 4529848320, 134217728;\n","recurrence_sentence":"The row polynomials satisfy (7*n-4+(8*n-1)*x+4*x^2)*P(n,x) = (42*n+18+(146*n+84)*x+(168*n+146)*x^2+(64*n+112)*x^3+32*x^4)*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[((42*n + 18 + (146*n + 84)*x + (168*n + 146)*x^2 + (64*n + 112)*x^3 + 32*x^4)*P[n - 1, x])/(7*n - 4 + (8*n - 1)*x + 4*x^2)]];\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] = ((42*n + 18 + (146*n + 84)*x + (168*n + 146)*x^2 + (64*n + 112)*x^3 + 32*x^4)*v[i - 1])/(7*n - 4 + (8*n - 1)*x + 4*x^2)); 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-eed4f744ce063213","steps":["basis A013609","D+1"],"edges":["e-185113cbc4b6c77b","e-9d563bdd96ef98c8"],"text":"A200536 \u2192 basis A013609 \u2192 D+1"},"headline":"A200536 \u2192 basis A013609 \u2192 D+1","chain":null}