{"id":"n-3d83195e5cd79843","revision":0,"first_index":0,"fingerprint":"f6-f03d443003d0524d","name":"mobius_x_over_x_plus_one 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":true,"rows":[["1"],["2","4"],["12","40","32"],["120","544","800","384"],["1680","9536","19904","18176","6144"],["30240","205248","548992","725248","474112","122880"],["665280","5241216","17003776","29131264","27836416","14082048","2949120"],["17297280","154899456","588852224","1233518592","1539422208","1145552896","470974464","82575360"],["518918400","5200496640","22622349312","55848828928","85655261184","83628097536","50786926592","17548443648","2642411520"],["17643225600","195497948160","956377755648","2713264758784","4922836721664","5926944718848","4737367015424","2424976441344","721585963008","95126814720"],["670442572800","8134651054080","44162272641024","141359171059712","295598385774592","422137368182784","417097735340032","281643785388032","124419324248064","32478257479680","3805072588800"],["28158588057600","371154817474560","2212740097302528","7880225180712960","18634935729160192","30736098437758976","36092149084323840","30181339341258752","17617938175492096","6838537081061376","1588856377835520","167423193907200"],["1295295050649600","18424733830594560","119601475714695168","468701882983317504","1235457852215590912","2308337443136602112","3135571582891327488","3120764765745446912","2259129444453056512","1160233738152443904","401337173271379968","83966573055836160","8036313307545600"],["64764752532480000","988592034163507200","6938159944925675520","29654370886330023936","86145390525912055808","179660680472040833024","276812126187669684224","319087960597621374976","275236240041416065024","175486418019181133824","80399013047496081408","25067861662903566336","4768077464280760320","417888291992371200"],["3497296636753920000","57010795986648268800","430021790939142881280","1989872984777659121664","6312495858033732354048","14525818614941280108544","25008852920568449597440","32732228938781056565248","32731682755703264509952","24889496015355080867840","14168786113638931890176","5856009260456573140992","1661276436193760772096","289578305767310622720","23401744351572787200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","84","1848","55440","2106720","96909120","5233092480","324451733760","22711621363200","1771506466329600","152349556104345600","14320858273808486400","1460727543928465612800","160680029832131217408000"],"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":[["-2","-2","0"],["4","8","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188703,"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-c89e01e8d0b897bb"]},{"id":188704,"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-c89e01e8d0b897bb"]},{"id":188705,"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-c89e01e8d0b897bb"]}],"region_exact":{},"incoming_edges":[{"id":"e-c89e01e8d0b897bb","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-3d83195e5cd79843","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":"2b145ca57536665287ba28e19da3234e7ee298aba2ac6f4f9d5c54bb1490908c","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-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-00eddef60c8f153b","kind":"fingerprint_related","node_a_id":"n-261c6497576cc5f7","node_b_id":"n-3d83195e5cd79843","fingerprint":"f6-f03d443003d0524d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f03d443003d0524d"},"other_id":"n-261c6497576cc5f7","other":{"id":"n-261c6497576cc5f7","name":"basis_map_A007318 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","first_index":0,"fingerprint":"f6-f03d443003d0524d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["8","10","3"]],"card_rows":[["1"],["2","1"],["8","10","3"],["48","100","68","15"]],"row_sums_preview":["1","3","21","231","3465","65835","1514205","40883535"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2b477f2aa77e3efc","steps":["basis A007318"],"edges":["e-07dfe9939e4be29b"],"text":"A109692 \u2192 basis A007318"},"headline":"A109692 \u2192 basis A007318","chain":null}},{"id":"l-1a01c1de2615bab0","kind":"fingerprint_related","node_a_id":"n-1e0cd3ca23b1a82f","node_b_id":"n-3d83195e5cd79843","fingerprint":"f6-f03d443003d0524d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f03d443003d0524d"},"other_id":"n-1e0cd3ca23b1a82f","other":{"id":"n-1e0cd3ca23b1a82f","name":"basis_map_A013609 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","first_index":0,"fingerprint":"f6-f03d443003d0524d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["3","10","8"]],"card_rows":[["1"],["1","2"],["3","10","8"],["15","68","100","48"]],"row_sums_preview":["1","3","21","231","3465","65835","1514205","40883535"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-849de7332df28657","steps":["basis A013609"],"edges":["e-0b105283f866e3e7"],"text":"A088996 \u2192 basis A013609"},"headline":"A088996 \u2192 basis A013609","chain":null}},{"id":"l-1cf3d27d4024c860","kind":"fingerprint_related","node_a_id":"n-3d83195e5cd79843","node_b_id":"n-d75f4aef92e17cce","fingerprint":"f6-f03d443003d0524d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f03d443003d0524d"},"other_id":"n-d75f4aef92e17cce","other":{"id":"n-d75f4aef92e17cce","name":"mobius_x_over_x_plus_one 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-f03d443003d0524d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["9","30","24"]],"card_rows":[["1"],["1","2"],["9","30","24"],["225","1020","1500","720"]],"row_sums_preview":["1","3","63","3465","363825","62214075","15740160975","5524796502225"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-857dad5fd144a15f","e-4df645d45cdacb8a"],"text":"A204420 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A204420 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","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,            4;\n  2:          12,           40,           32;\n  3:         120,          544,          800,           384;\n  4:        1680,         9536,        19904,         18176,          6144;\n  5:       30240,       205248,       548992,        725248,        474112,        122880;\n  6:      665280,      5241216,     17003776,      29131264,      27836416,      14082048,       2949120;\n  7:    17297280,    154899456,    588852224,    1233518592,    1539422208,    1145552896,     470974464,      82575360;\n  8:   518918400,   5200496640,  22622349312,   55848828928,   85655261184,   83628097536,   50786926592,   17548443648,   2642411520;\n  9: 17643225600, 195497948160, 956377755648, 2713264758784, 4922836721664, 5926944718848, 4737367015424, 2424976441344, 721585963008, 95126814720;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (4*n-2+(8*n-2)*x+4*n*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[((4*n - 2 + (8*n - 2)*x + 4*n*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] = ((4*n - 2 + (8*n - 2)*x + 4*n*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-4172472abe953cea","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-294a6ed01a588998","e-c89e01e8d0b897bb"],"text":"A290319 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A290319 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}
