{"id":"n-ed5fd6fb94c974b5","revision":1,"first_index":0,"fingerprint":"f6-7d5966053f53b1d0","name":"basis_map_A008277 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","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":false,"rows":[["1"],["1","1"],["9","15","3"],["225","495","180","15"],["11025","29295","14805","2310","105"],["893025","2735775","1729350","382725","33075","945"],["108056025","370945575","277390575","77962500","9667350","530145","10395"],["18261468225","68916822975","58913454600","19822277475","3149996850","249729480","9459450","135135"],["4108830350625","16813959537375","16056683267625","6235716737250","1195083264375","122168796750","6725668950","186486300","2027025"],["1187451971330625","5214921734397375","5472146007578250","2393107306214625","532651909915125","66007545478875","4706606104200","190422782550","4031752725","34459425"],["428670161650355625","2004231846526284375","2282469769977271875","1104289628141700000","278237923450987500","40219827026754375","3489781263593625","183494370559500","5686322016375","94935715875","654729075"],["189043541287806830625","934957186489800849375","1144521767676074287500","604493837752939509375","169264992002744137500","27803021949078626250","2823322344019427250","181180096424452125","7316351889721875","179222263345125","2419878661200","13749310575"],["100004033341249813400625","520444368391989625959375","679604031703008134848125","387811315632807535218750","119016918302021718103125","21798240951269750467500","2522625508289766379500","189950680445318793000","9383701115134095750","300183679285615125","5958799960788675","66409170077250","316234143225"],["62502520838281133375390625","340788940288324502208609375","471743830609330231671468750","288458021735540478024328125","95982548659785934005234375","19324103671002855803203125","2499990792706312705800000","215026182063499193887500","12490320431118598818750","489748655271125250000","12718541947830468750","208738252089241875","1952745834414375","7905853580625"],["45564337691106946230659765625","259324006920606914270844234375","378576720754890332958392484375","246388081032239703454835437500","88125759815100419669038031250","19287554385560687632108828125","2749242595418401727554284375","264893786537362703045325000","17609059723121554118681250","813274344563735911106250","25929037098203020031250","557566079234984820000","7692174170047867500","61262459396263125","213458046676875"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","27","915","57540","5774895","844562565","169323344190","44539356136275","14871187570013775","6141798214319968200","3073096347606669158775","1831401174421401447890025","1281527975775362188319696250","1040299045809885281462003836875"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-1"],["-4","2"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1"],["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166178,"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-86d697d3c8220248"]},{"id":166179,"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-86d697d3c8220248"]},{"id":166180,"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-86d697d3c8220248"]}],"region_exact":{},"incoming_edges":[{"id":"e-86d697d3c8220248","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-ed5fd6fb94c974b5","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"001b78caee35522dce87a813d07e117660639f424aa71dc403138053e47ed7bc","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-1336d058dca9979e","role":"sequence","name":"Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..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:                1,                1;\n  2:                9,               15,                3;\n  3:              225,              495,              180,               15;\n  4:            11025,            29295,            14805,             2310,             105;\n  5:           893025,          2735775,          1729350,           382725,           33075,            945;\n  6:        108056025,        370945575,        277390575,         77962500,         9667350,         530145,         10395;\n  7:      18261468225,      68916822975,      58913454600,      19822277475,      3149996850,      249729480,       9459450,       135135;\n  8:    4108830350625,   16813959537375,   16056683267625,    6235716737250,   1195083264375,   122168796750,    6725668950,    186486300,    2027025;\n  9: 1187451971330625, 5214921734397375, 5472146007578250, 2393107306214625, 532651909915125, 66007545478875, 4706606104200, 190422782550, 4031752725, 34459425;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n^2-4*n+1+(2*n-1)*x)*P(n-1,x) + (2*n-1)*x*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[(4*n^2 - 4*n + 1 + (2*n - 1)*x)*P[n - 1, x] + (2*n - 1)*x*D[P[n - 1, x], 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] = (4*n^2 - 4*n + 1 + (2*n - 1)*x)*v[i - 1] + (2*n - 1)*x*deriv(v[i - 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-1336d058dca9979e","steps":["basis A008277"],"edges":["e-86d697d3c8220248"],"text":"A204420 \u2192 basis A008277"},"headline":"A204420 \u2192 basis A008277","chain":null}