{"id":"n-1c1963609d82ab65","revision":1,"first_index":0,"fingerprint":"f6-68e273231fc13fce","name":"basis_map_A007318 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","1"],["12","8","1"],["120","92","18","1"],["1680","1408","344","32","1"],["30240","27024","7600","920","50","1"],["665280","624768","194224","27840","2020","72","1"],["17297280","16909248","5674592","918064","80360","3892","98","1"],["518918400","524574720","187147008","33216512","3328864","197120","6832","128","1"],["17643225600","18354458880","6887572992","1316508416","146397888","10030944","429408","11184","162","1"],["670442572800","715112663040","280082232576","56914892800","6879628160","527573760","26348448","854400","17340","200","1"],["28158588057600","30705174420480","12478566431232","2670507730176","345859275520","29037726080","1634208576","62233248","1582680","25740","242","1"],["1295295050649600","1440596611399680","604719230257152","135321922019328","18580034404096","1681594675200","104211320576","4496937984","135036528","2766720","36872","288","1"],["64764752532480000","73325125620633600","31676558124257280","7370815331223552","1064323642224128","102659768164096","6892160704000","329058219776","11248764384","273372528","4610320","51272","338","1"],["3497296636753920000","4024321536046694400","1783859264330526720","429700586010329088","64844292011326464","6607951123085312","474836446180096","24661304571904","936491496512","26010880896","522329808","7379008","69524","392","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","21","231","3465","65835","1514205","40883535","1267389585","44358635475","1729986783525","74389431691575","3496303289504025","178311467764705275","9807130727058790125"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2","1"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161981,"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-741861c2575e0643"]},{"id":161982,"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-741861c2575e0643"]},{"id":161983,"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-741861c2575e0643"]}],"region_exact":{},"incoming_edges":[{"id":"e-741861c2575e0643","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-1c1963609d82ab65","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"294ba3e7653c8dd5389eecfd2ab0c0c3456db4b7ce62a5bdc7cede32181bf3f7","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-4172472abe953cea","role":"sequence","name":"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-ce1e44acd41ebf11","kind":"oeis_match","node_a_id":"n-1c1963609d82ab65","node_b_id":null,"fingerprint":null,"oeis_id":"A008545","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A008545","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A008545","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-1c1963609d82ab65","oeis_id":"A008545","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":18,"oeis_terms_available":18,"term_count":200,"raw_size":45930}],"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:          12,           8,          1;\n  3:         120,          92,         18,          1;\n  4:        1680,        1408,        344,         32,         1;\n  5:       30240,       27024,       7600,        920,        50,        1;\n  6:      665280,      624768,     194224,      27840,      2020,       72,      1;\n  7:    17297280,    16909248,    5674592,     918064,     80360,     3892,     98,     1;\n  8:   518918400,   524574720,  187147008,   33216512,   3328864,   197120,   6832,   128,   1;\n  9: 17643225600, 18354458880, 6887572992, 1316508416, 146397888, 10030944, 429408, 11184, 162, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n-2+x)*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]];\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]); 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 A007318"],"edges":["e-741861c2575e0643"],"text":"A290319 \u2192 basis A007318"},"headline":"A290319 \u2192 basis A007318","chain":null}