{"id":"n-bd5cee0018a9f607","revision":0,"first_index":0,"fingerprint":"f6-ff79ef52271940c5","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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":true,"rows":[["1"],["3","7"],["15","74","91"],["105","803","2043","1729"],["945","9852","38462","66636","43225"],["10395","137667","728494","1925318","2541191","1339975"],["135135","2174286","14564101","51983412","104272249","111443742","49579075"],["2027025","38425095","311955813","1406007523","3799370451","6155362837","5535767031","2131900225"],["34459425","752550840","7186078476","39187962728","133483666294","290810320328","395720818540","307494888344","104463111025"],["654729075","16193734335","177925787244","1139805608012","4691527609626","12866997732402","23513263170300","27607047898236","18897017968395","5745471106375"],["13749310575","380006894610","4724259326559","34789390790136","168050221890878","556390136567628","1278665388252822","2014057059251256","2080867299128691","1273372989305970","350473737488875"],["316234143225","9661362384555","134118426449727","1116681363052581","6196044286429306","24056338007744270","66687443079845982","131993893375717962","182801770849794045","168705687795659607","93376886245744115","23481740411754625"],["7905853580625","264619152069300","4058240115315690","37707679207144596","236418846663571063","1053719683102946088","3423298751561481260","8168030679221705736","14205598487672262351","17562171466926455460","14649937365226754186","7403556206233186020","1714167050058087625"],["213458046676875","7769279538740475","130477396126998330","1338708307702843602","9362215517280841785","47127520330201658353","175672921257292734972","490977429712343074412","1028825582825665836621","1596420910133123023149","1782959854748312344362","1357241069421209603234","631163450643990061455","135419196954588922375"],["6190283353629375","243453040591008150","4444233248475891945","49913119594177322508","385294456155886117935","2162486408544719643962","9100353944528630274193","29170543768527831630568","71569023427493470587029","133746380934042163784106","187401613149016514954163","190911578666821627764556","133669230969478528057085","57576050016422233972550","11510631741140058401875"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","180","4680","159120","6683040","334152000","19380816000","1279133856000","94655905344000","7761784238208000","698560581438720000","68458936980994560000","7256647319985423360000","827257794478338263040000"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["1","2","1"],["2","8","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191521,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-079f0c412fff8a5b"]},{"id":191522,"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-079f0c412fff8a5b"]},{"id":191523,"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-079f0c412fff8a5b"]}],"region_exact":{},"incoming_edges":[{"id":"e-079f0c412fff8a5b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-bd5cee0018a9f607","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"4653ffe9e69cbd9aece20bc1148d2d224018d4eac3734a26381f4a96d9413ed9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-be7df2e3fd154f3c","role":"sequence","name":"basis_map_A013611 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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,           7;\n  2:        15,          74,           91;\n  3:       105,         803,         2043,          1729;\n  4:       945,        9852,        38462,         66636,         43225;\n  5:     10395,      137667,       728494,       1925318,       2541191,        1339975;\n  6:    135135,     2174286,     14564101,      51983412,     104272249,      111443742,       49579075;\n  7:   2027025,    38425095,    311955813,    1406007523,    3799370451,     6155362837,     5535767031,     2131900225;\n  8:  34459425,   752550840,   7186078476,   39187962728,  133483666294,   290810320328,   395720818540,   307494888344,   104463111025;\n  9: 654729075, 16193734335, 177925787244, 1139805608012, 4691527609626, 12866997732402, 23513263170300, 27607047898236, 18897017968395, 5745471106375;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (2*n+1+(8*n+2)*x+(6*n+1)*x^2)*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[((2*n + 1 + (8*n + 2)*x + (6*n + 1)*x^2)*P[n - 1, x])/(1 + x)]];\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] = ((2*n + 1 + (8*n + 2)*x + (6*n + 1)*x^2)*v[i - 1])/(1 + x)); 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-7fd16a32c438d522","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-5d9e378754634336","e-079f0c412fff8a5b"],"text":"A059364 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A059364 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}
