{"id":"n-20aa41dfc4e18a9a","revision":0,"first_index":0,"fingerprint":"f6-7d048716a74a4e37","name":"basis_map_A008277 of basis_map_A007318 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"],["2","1"],["24","21","3"],["720","855","225","15"],["40320","58905","21735","2730","105"],["3628800","6194475","2877525","515025","37800","945"],["479001600","925726725","511278075","116631900","12318075","592515","10395"],["87178291200","186743732175","118380287025","32422264875","4398644250","306486180","10405395","135135"],["20922789888000","48927326072625","34763833479375","11016049794750","1805927248125","162522810450","8031073050","202702500","2027025"],["6402373705728000","16159213749553875","12651467926222125","4523714945875125","862689637309500","94247182104075","6039565582050","222676804350","4341887550","34459425"],["2432902008176640000","6569171386480828125","5595338654402371875","2217241321945650000","479337058584759375","61158514608316125","4774241857510125","228984946690500","6540743459250","101483006625","654729075"],["1124000727777607680000","3224000721841949424375","2958003280934892815625","1281553805763916059375","308280101748137268750","44742956013195189750","4091583018990592125","239710911542227125","8929352259828000","203421049957125","2571121077525","13749310575"],["620448401733239439360000","1879652431798005895655625","1843037978601430740504375","863878988840894407631250","228008123058370470440625","36933994001008348744500","3852280271406997930500","265020289366391559000","12085354228704631875","359771678893501875","6689300831638425","70203979795950","316234143225"],["403291461126605635584000000","1284276601506984965551546875","1337117895815951665744453125","672388216374684214045265625","192603067014800213021250000","34324048427240732038003125","4005174067150807063012500","314948745512447984437500","16898058328558726068750","616934074749429937500","15014662720812129375","232171202102214375","2055521930962500","7905853580625"],["304888344611713860501504000000","1016477448430387580187629203125","1117740963851609443322898796875","598891120292641382130987562500","184563531742903857829582171875","35783009958071097997434534375","4603499101179940648871559375","405766264322335135994775000","24928528824195177318637500","1072564715545766111418750","32062263969787853051250","649872169275559230000","8488586142199288125","64250872049739375","213458046676875"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","48","1815","123795","13254570","2045559285","429440246235","117606685095900","40699973765526075","17360158813883961075","8944922023368411430350","5476089670479451335537225","3928338943320231339886861875","3263379718067058428234040645000"],"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":[["0","-1"],["-2","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":198064,"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-87dc5132438f4e40"]},{"id":198065,"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-87dc5132438f4e40"]},{"id":198066,"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-87dc5132438f4e40"]}],"region_exact":{},"incoming_edges":[{"id":"e-87dc5132438f4e40","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-20aa41dfc4e18a9a","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":"d491ed92aaa0dce6e51cd7590e7d091d4d3047eda239553318a17d4bbf5f6110","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-2da216d37409bd5d","role":"sequence","name":"basis_map_A007318 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-32e4b12dd1c7737e","kind":"fingerprint_related","node_a_id":"n-4aa4785d70eabbb3","node_b_id":"n-20aa41dfc4e18a9a","fingerprint":"f6-7d048716a74a4e37","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7d048716a74a4e37"},"other_id":"n-4aa4785d70eabbb3","other":{"id":"n-4aa4785d70eabbb3","name":"basis_map_A008277 of basis_map_A013609 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":"f6-7d048716a74a4e37","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["32","28","4"]],"card_rows":[["1"],["4","2"],["32","28","4"],["384","456","120","8"]],"row_sums_preview":["1","6","64","968","18864","448832","12594112","406766208"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4172472abe953cea","steps":["basis A013609","basis A008277"],"edges":["e-294a6ed01a588998","e-eabee376b52d8590"],"text":"A290319 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A290319 \u2192 basis A013609 \u2192 basis A008277","chain":null}},{"id":"l-770a5b5e02f5ac89","kind":"fingerprint_related","node_a_id":"n-d7b6e6dc2dbf6491","node_b_id":"n-20aa41dfc4e18a9a","fingerprint":"f6-7d048716a74a4e37","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7d048716a74a4e37"},"other_id":"n-d7b6e6dc2dbf6491","other":{"id":"n-d7b6e6dc2dbf6491","name":"basis_map_A008277 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-7d048716a74a4e37","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["8","7","1"]],"card_rows":[["1"],["2","1"],["8","7","1"],["48","57","15","1"]],"row_sums_preview":["1","3","16","121","1179","14026","196783","3177861"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A008277"],"edges":["e-760cba6118ea84eb"],"text":"A028338 \u2192 basis A008277"},"headline":"A028338 \u2192 basis A008277","chain":null}},{"id":"l-db97b05fe6ca3d47","kind":"fingerprint_related","node_a_id":"n-8e129ac4447a70d3","node_b_id":"n-20aa41dfc4e18a9a","fingerprint":"f6-7d048716a74a4e37","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7d048716a74a4e37"},"other_id":"n-8e129ac4447a70d3","other":{"id":"n-8e129ac4447a70d3","name":"reversal of basis_map_A008277 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-7d048716a74a4e37","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","7","8"]],"card_rows":[["1"],["1","2"],["1","7","8"],["1","15","57","48"]],"row_sums_preview":["1","3","16","121","1179","14026","196783","3177861"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A008277","reverse"],"edges":["e-760cba6118ea84eb","e-d969d9961c45ee50"],"text":"A028338 \u2192 basis A008277 \u2192 reverse"},"headline":"A028338 \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:                2,                 1;\n  2:               24,                21,                 3;\n  3:              720,               855,               225,               15;\n  4:            40320,             58905,             21735,             2730,             105;\n  5:          3628800,           6194475,           2877525,           515025,           37800,            945;\n  6:        479001600,         925726725,         511278075,        116631900,        12318075,         592515,         10395;\n  7:      87178291200,      186743732175,      118380287025,      32422264875,      4398644250,      306486180,      10405395,       135135;\n  8:   20922789888000,    48927326072625,    34763833479375,   11016049794750,   1805927248125,   162522810450,    8031073050,    202702500,    2027025;\n  9: 6402373705728000, 16159213749553875, 12651467926222125, 4523714945875125, 862689637309500, 94247182104075, 6039565582050, 222676804350, 4341887550, 34459425;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n^2-2*n+(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 - 2*n + (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 - 2*n + (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 A007318","basis A008277"],"edges":["e-857dad5fd144a15f","e-87dc5132438f4e40"],"text":"A204420 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A204420 \u2192 basis A007318 \u2192 basis A008277","chain":null}