{"id":"n-837649989bd08cfc","revision":0,"first_index":0,"fingerprint":"f6-81eb69734bdba80b","name":"basis_map_A007318 of Pascal (1,6) 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":[["6"],["7","6"],["14","19","6"],["28","52","31","6"],["56","132","114","43","6"],["112","320","360","200","55","6"],["224","752","1040","760","310","67","6"],["448","1728","2832","2560","1380","444","79","6"],["896","3904","7392","7952","5320","2268","602","91","6"],["1792","8704","18688","23296","18592","9856","3472","784","103","6"],["3584","19200","46080","65280","60480","38304","16800","5040","990","115","6"],["7168","41984","111360","176640","186240","137088","71904","26880","7020","1220","127","6"],["14336","91136","264704","464640","549120","460416","280896","125664","40920","9460","1474","139","6"],["28672","196608","620544","1193984","1562880","1469952","1022208","532224","207504","59840","12408","1752","151","6"],["57344","421888","1437696","3008512","4319744","4502784","3514368","2086656","947232","327184","84656","15912","2054","163","6"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["6","13","39","117","351","1053","3159","9477","28431","85293","255879","767637","2302911","6908733","20726199"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["6"],["7","6"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195256,"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-89b1c48a1bc6db2c"]},{"id":195257,"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-89b1c48a1bc6db2c"]},{"id":195258,"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-89b1c48a1bc6db2c"]},{"id":195259,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-89b1c48a1bc6db2c"]}],"region_exact":{},"incoming_edges":[{"id":"e-89b1c48a1bc6db2c","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-837649989bd08cfc","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":"feab22933c1cfe6affa4f24af1562a0dc4a1cce582f5f6229bbaa990465a9d4d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b58d0e199d27164c","role":"sequence","name":"Pascal (1,6) 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":[],"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:    6;\n  1:    7,    6;\n  2:   14,   19,     6;\n  3:   28,   52,    31,     6;\n  4:   56,  132,   114,    43,     6;\n  5:  112,  320,   360,   200,    55,    6;\n  6:  224,  752,  1040,   760,   310,   67,    6;\n  7:  448, 1728,  2832,  2560,  1380,  444,   79,   6;\n  8:  896, 3904,  7392,  7952,  5320, 2268,  602,  91,   6;\n  9: 1792, 8704, 18688, 23296, 18592, 9856, 3472, 784, 103, 6;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 6, P(1,x) = 7 + 6*x.","mathematica":"P[0, x_] := 6;\nP[1, x_] := 7 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 6; v[2] = 7 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*x)*v[i - 1])/(n - 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-b58d0e199d27164c","steps":["basis A007318"],"edges":["e-89b1c48a1bc6db2c"],"text":"A096956 \u2192 basis A007318"},"headline":"A096956 \u2192 basis A007318","chain":null}
