{"id":"n-2ac02caff5f913e9","revision":0,"first_index":0,"fingerprint":"f6-37a16ca70b47038c","name":"basis_map_A008277 of basis_map_A013609 of Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["4","2"],["10","6"],["46","60","12"],["166","260","60"],["856","2220","1080","120"],["3844","12012","6720","840"],["21820","101696","90720","23520","1680"],["114076","643680","659232","191520","15120"],["703216","5612760","8205120","3729600","604800","30240"],["4125496","40078280","67224960","34151040","6098400","332640"],["27331624","364250832","822499920","583894080","161330400","17962560","665280"],["175849480","2867959536","7431275280","5885605440","1785944160","216216000","8648640"],["1241782816","27254885840","91749289632","95586210720","40037437440","7446479040","605404800","17297280"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","6","16","118","486","4276","23416","239436","1623628","18885736","152010816","1977934696","18371498536","263938787568"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","-2"],["3","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-2"],["0","2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196489,"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-7da92287d234c568"]},{"id":196490,"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-7da92287d234c568"]},{"id":196491,"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-7da92287d234c568"]}],"region_exact":{},"incoming_edges":[{"id":"e-7da92287d234c568","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-2ac02caff5f913e9","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":"a7faa3f3d52eaeeb2d640a67afa263c4db2001e8510c09fbd0d0dc3f127b97a4","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-6a6677e3386580cd","role":"sequence","name":"basis_map_A013609 of Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(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":[],"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:      1;\n  2:      4,      2;\n  3:     10,      6;\n  4:     46,     60,     12;\n  5:    166,    260,     60;\n  6:    856,   2220,   1080,    120;\n  7:   3844,  12012,   6720,    840;\n  8:  21820, 101696,  90720,  23520,  1680;\n  9: 114076, 643680, 659232, 191520, 15120;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-3+(2*n-2)*x)*P(n-2,x) + (2*n-2)*x*d/dx P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n - 3 + (2*n - 2)*x)*P[n - 2, x] + (2*n - 2)*x*D[P[n - 2, x], 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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (3*n - 3 + (2*n - 2)*x)*v[i - 2] + (2*n - 2)*x*deriv(v[i - 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-97b51734d39ede6c","steps":["basis A013609","basis A008277"],"edges":["e-4139d19ff9e22325","e-7da92287d234c568"],"text":"A100861 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A100861 \u2192 basis A013609 \u2192 basis A008277","chain":null}
