{"id":"n-8090c1757ab2328b","revision":0,"first_index":0,"fingerprint":"f6-357ed7a70cf3dab3","name":"basis_map_A008277 of basis_map_A013609 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"],["3","2"],["45","60","12"],["1575","3150","1260","120"],["99225","264600","158760","30240","1680"],["9823275","32744250","26195400","7484400","831600","30240"],["1404728325","5618913300","5618913300","2140538400","356756400","25945920","665280"],["273922023375","1278302775750","1533963330900","730458729000","162324162000","17708090400","908107200","17297280"],["69850115960625","372533951790000","521547532506000","298027161432000","82785322620000","12041501472000","926269344000","35286451200","518918400"],["22561587455281875","135369524731691250","216591239570706000","144394159713804000","48131386571268000","8751161194776000","897554994336000","51288856819200","1508495788800","17643225600"],["9002073394657468125","60013822631049787500","108024880735889617500","82304671036868280000","32007372069893220000","6983426633431248000","895311106850160000","68214179569536000","3009449098656000","70396470144000","670442572800"],["4348001449619557104375","31885343963876752098750","63770687927753504197500","54660589652360146455000","24293595401048953980000","6183824283903370104000","951357582138980016000","90605484013236192000","5329734353719776000","187008222937536000","3562061389286400","28158588057600"],["2500100833531245335015625","20000806668249962680125000","44001774670149917896275000","41906452066809445615500000","20953226033404722807750000","6095483936990464816800000","1094061219459827018400000","125035567938265944960000","9193791760166613600000","430118912756332800000","12289111793038080000","194294257597440000","1295295050649600"],["1687568062633590601135546875","14625589876157785209841406250","35101415702778684503619375000","36772911688625288527601250000","20429395382569604737556250000","6685983943386416095927500000","1371483885822854583780000000","182864518109713944504000000","16135104539092406868000000","943573364859205080000000","35945651994636384000000","852466055208768000000","11366214069450240000","64764752532480000"],["1321365793042101440689133203125","12332747401726280113098576562500","32065143244488328294056299062500","36645877993700946621778627500000","22394703218372800713309161250000","8143528443044654804839695000000","1879275794548766493424545000000","286365835359812037093264000000","29478835992921827347836000000","2068690245117321217392000000","98509059291301010352000000","3114910965732838272000000","62298219314656765440000","709951217261045760000","3497296636753920000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","117","6105","554505","77109165","15166460925","3997604515905","1357747660494225","576749429227696725","299302852304220689925","186189526077511134803625","136686577391908120747395225","116874328547536918483048048125","115100656937503654529257718698125"],"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","-2"],["0","4"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2"],["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":198112,"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-bc3c6ed364a44e85"]},{"id":198113,"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-bc3c6ed364a44e85"]},{"id":198114,"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-bc3c6ed364a44e85"]}],"region_exact":{},"incoming_edges":[{"id":"e-bc3c6ed364a44e85","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-8090c1757ab2328b","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":"ef91c905434948d485799422266fea49fff839b0893b6a6847cc483eb00b5365","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-22e887254e1ffff8","role":"sequence","name":"basis_map_A013609 of 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":[{"id":"l-05ae2c904958cfe1","kind":"fingerprint_related","node_a_id":"n-42e8691f6ca17cbd","node_b_id":"n-8090c1757ab2328b","fingerprint":"f6-357ed7a70cf3dab3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-357ed7a70cf3dab3"},"other_id":"n-42e8691f6ca17cbd","other":{"id":"n-42e8691f6ca17cbd","name":"basis_map_A008277 of basis_map_A013611 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","first_index":0,"fingerprint":"f6-357ed7a70cf3dab3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["60","80","16"]],"card_rows":[["1"],["6","4"],["60","80","16"],["840","1680","672","64"]],"row_sums_preview":["1","10","156","3256","84496","2611104","93376960","3786534784"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4172472abe953cea","steps":["basis A013611","basis A008277"],"edges":["e-403803ced55fc794","e-cf73544ec8921c50"],"text":"A290319 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A290319 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-086c7492cfe1ecf2","kind":"fingerprint_related","node_a_id":"n-deb68900633836aa","node_b_id":"n-8090c1757ab2328b","fingerprint":"f6-357ed7a70cf3dab3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-357ed7a70cf3dab3"},"other_id":"n-deb68900633836aa","other":{"id":"n-deb68900633836aa","name":"basis_map_A008277 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","first_index":1,"fingerprint":"f6-357ed7a70cf3dab3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["15","20","4"]],"card_rows":[["1"],["3","2"],["15","20","4"],["105","210","84","8"]],"row_sums_preview":["1","5","39","407","5281","81597","1459015","29582303"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c9c430277778c10","steps":["basis A008277"],"edges":["e-0664c5d711618d77"],"text":"A161198 \u2192 basis A008277"},"headline":"A161198 \u2192 basis A008277","chain":null}},{"id":"l-0b1799bef0daeda2","kind":"fingerprint_related","node_a_id":"n-80bb1878a7313c9f","node_b_id":"n-8090c1757ab2328b","fingerprint":"f6-357ed7a70cf3dab3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-357ed7a70cf3dab3"},"other_id":"n-80bb1878a7313c9f","other":{"id":"n-80bb1878a7313c9f","name":"reversal of basis_map_A008277 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","first_index":1,"fingerprint":"f6-357ed7a70cf3dab3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["4","20","15"]],"card_rows":[["1"],["2","3"],["4","20","15"],["8","84","210","105"]],"row_sums_preview":["1","5","39","407","5281","81597","1459015","29582303"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c9c430277778c10","steps":["basis A008277","reverse"],"edges":["e-0664c5d711618d77","e-41f6ca9a401302a7"],"text":"A161198 \u2192 basis A008277 \u2192 reverse"},"headline":"A161198 \u2192 basis A008277 \u2192 reverse","chain":null}}],"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,                  2;\n  2:                45,                 60,                 12;\n  3:              1575,               3150,               1260,                120;\n  4:             99225,             264600,             158760,              30240,              1680;\n  5:           9823275,           32744250,           26195400,            7484400,            831600,            30240;\n  6:        1404728325,         5618913300,         5618913300,         2140538400,         356756400,         25945920,          665280;\n  7:      273922023375,      1278302775750,      1533963330900,       730458729000,      162324162000,      17708090400,       908107200,       17297280;\n  8:    69850115960625,    372533951790000,    521547532506000,    298027161432000,    82785322620000,   12041501472000,    926269344000,    35286451200,     518918400;\n  9: 22561587455281875, 135369524731691250, 216591239570706000, 144394159713804000, 48131386571268000, 8751161194776000, 897554994336000, 51288856819200, 1508495788800, 17643225600;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n^2-1+(4*n-2)*x)*P(n-1,x) + (4*n-2)*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 - 1 + (4*n - 2)*x)*P[n - 1, x] + (4*n - 2)*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 - 1 + (4*n - 2)*x)*v[i - 1] + (4*n - 2)*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 A013609","basis A008277"],"edges":["e-2613ba1fbc2604f1","e-bc3c6ed364a44e85"],"text":"A204420 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A204420 \u2192 basis A013609 \u2192 basis A008277","chain":null}