{"id":"n-4aa4785d70eabbb3","revision":0,"first_index":0,"fingerprint":"f6-7d048716a74a4e37","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.","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"],["4","2"],["32","28","4"],["384","456","120","8"],["6144","8976","3312","416","16"],["122880","209760","97440","17440","1280","32"],["2949120","5699520","3147840","718080","75840","3648","64"],["82575360","176883840","112129920","30710400","4166400","290304","9856","128"],["2642411520","6179201280","4390444800","1391255040","228076800","20525568","1014272","25600","256"],["95126814720","240094471680","187976194560","67213601280","12817889280","1400329728","89736192","3308544","64512","512"],["3805072588800","10274221440000","8751141504000","3467778048000","749685888000","95652264960","7466941440","358133760","10229760","158720","1024"],["167423193907200","480224331417600","440603235072000","190891185408000","45919222272000","6664594083840","609453250560","35705640960","1330053120","30300160","382976","2048"],["8036313307545600","24346062958694400","23871816886579200","11189330482176000","2953258818048000","478384901406720","49896383201280","3432656240640","156534681600","4659916800","86642688","909312","4096"],["417888291992371200","1330760026384588800","1385513871565824000","696724801739366400","199574190047232000","35566381523312640","4150138327695360","326348078284800","17509670830080","639263539200","15558107136","240574464","2129920","8192"],["23401744351572787200","78019858114290892800","85792352346718617600","45967965450972364800","14166197766501580800","2746529560633098240","353342169329909760","31144642135326720","1913392456335360","82324843560960","2460943220736","49881022464","651542528","4931584","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","64","968","18864","448832","12594112","406766208","14852955136","604722411008","27151387198464","1332372281790464","70928657676046336","4070522214393864192","250481132630935207936"],"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","2"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197698,"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-eabee376b52d8590"]},{"id":197699,"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-eabee376b52d8590"]},{"id":197700,"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-eabee376b52d8590"]}],"region_exact":{},"incoming_edges":[{"id":"e-eabee376b52d8590","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-4aa4785d70eabbb3","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":"a5b91991b939895ac230d57e6d43a9954ad07d2857e3f66480d64c2f95e9c9d5","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-de50e58c7c589630","role":"sequence","name":"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.","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-1ae3ae27cd7048aa","kind":"fingerprint_related","node_a_id":"n-8e129ac4447a70d3","node_b_id":"n-4aa4785d70eabbb3","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}},{"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-20aa41dfc4e18a9a","other":{"id":"n-20aa41dfc4e18a9a","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.","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"],["24","21","3"]],"card_rows":[["1"],["2","1"],["24","21","3"],["720","855","225","15"]],"row_sums_preview":["1","3","48","1815","123795","13254570","2045559285","429440246235"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-3b12113c9b6d9ac0","kind":"fingerprint_related","node_a_id":"n-d7b6e6dc2dbf6491","node_b_id":"n-4aa4785d70eabbb3","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}}],"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:           4,            2;\n  2:          32,           28,            4;\n  3:         384,          456,          120,           8;\n  4:        6144,         8976,         3312,         416,          16;\n  5:      122880,       209760,        97440,       17440,        1280,         32;\n  6:     2949120,      5699520,      3147840,      718080,       75840,       3648,       64;\n  7:    82575360,    176883840,    112129920,    30710400,     4166400,     290304,     9856,     128;\n  8:  2642411520,   6179201280,   4390444800,  1391255040,   228076800,   20525568,  1014272,   25600,   256;\n  9: 95126814720, 240094471680, 187976194560, 67213601280, 12817889280, 1400329728, 89736192, 3308544, 64512, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+2*x)*P(n-1,x) + 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*x)*P[n - 1, x] + 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*x)*v[i - 1] + 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-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}
