{"id":"n-cfaeb7e6453df3d4","revision":0,"first_index":0,"fingerprint":"f6-d7e937c0efe9d85c","name":"basis_map_A007318 of Elements in 3-Pascal triangle (by row).","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"],["5","5","1"],["10","15","7","1"],["20","40","29","9","1"],["40","100","98","47","11","1"],["80","240","296","192","69","13","1"],["160","560","832","680","330","95","15","1"],["320","1280","2224","2192","1340","520","125","17","1"],["640","2880","5728","6608","4872","2380","770","159","19","1"],["1280","6400","14336","18944","16352","9632","3920","1088","197","21","1"],["2560","14080","35072","52224","51648","35616","17472","6096","1482","239","23","1"],["5120","30720","84224","139520","155520","122880","70560","29664","9060","1960","285","25","1"],["10240","66560","199168","363264","450560","401280","264000","129888","47784","12980","2530","335","27","1"],["20480","143360","464896","925696","1264384","1253120","929280","523776","225456","73744","18040","3200","389","29","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","11","33","99","297","891","2673","8019","24057","72171","216513","649539","1948617","5845851"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"],["5","5","1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","-2"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195204,"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-5e42a151ff6701f1"]},{"id":195205,"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-5e42a151ff6701f1"]},{"id":195206,"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-5e42a151ff6701f1"]},{"id":195207,"position":3,"claim":"interlacing","from_index":3,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 3","region_exact":null,"edges":["e-5e42a151ff6701f1"]}],"region_exact":{},"incoming_edges":[{"id":"e-5e42a151ff6701f1","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-cfaeb7e6453df3d4","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":"983b99ebd1a700ab58197db9e0da9ecc6f500bdb3eef4969af44c572232c03f5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-347cb2d3d9dd489a","role":"sequence","name":"Elements in 3-Pascal triangle (by row).","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:   2,    1;\n  2:   5,    5,    1;\n  3:  10,   15,    7,    1;\n  4:  20,   40,   29,    9,    1;\n  5:  40,  100,   98,   47,   11,    1;\n  6:  80,  240,  296,  192,   69,   13,   1;\n  7: 160,  560,  832,  680,  330,   95,  15,   1;\n  8: 320, 1280, 2224, 2192, 1340,  520, 125,  17,  1;\n  9: 640, 2880, 5728, 6608, 4872, 2380, 770, 159, 19, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 5 + 5*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 5 + 5*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (n - 2)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 5 + 5*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (n - 2)*x)*v[i - 1])/(n - 2)); 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-347cb2d3d9dd489a","steps":["basis A007318"],"edges":["e-5e42a151ff6701f1"],"text":"A028262 \u2192 basis A007318"},"headline":"A028262 \u2192 basis A007318","chain":null}
