{"id":"n-bc12291099a8ab59","revision":0,"first_index":0,"fingerprint":"f6-7f5e6b8bb14328be","name":"basis_map_A013609 of A Motzkin related 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":false,"rows":[["1"],["1","2"],["2","6","4"],["4","18","24","8"],["9","52","104","80","16"],["21","150","400","480","240","32"],["51","432","1440","2360","1920","672","64"],["127","1246","4984","10360","11760","6944","1792","128"],["323","3600","16800","42336","61824","51968","23296","4608","256"],["835","10422","55584","164640","294336","322560","209664","73728","11520","512"],["2188","30230","181380","617520","1307040","1770048","1518720","787200","222720","28160","1024"],["5798","87846","585640","2253240","5512320","8899968","9520896","6589440","2787840","647680","67584","2048"],["15511","255708","1875192","8045840","22342320","41918976","53822208","46835712","26737920","9405440","1824768","159744","4096"],["41835","745498","5963984","28233920","87747088","187675488","281478912","295865856","214046976","102575616","30457856","5005312","372736","8192"],["113634","2176566","18863572","97668480","335903568","806886080","1385319936","1709465472","1502941440","919446528","375158784","95234048","13418496","860160","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","12","54","261","1323","6939","37341","205011","1143801","6466230","36960300","213243435","1240219269","7263473148"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","2"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-3","-4","4"],["3","4","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":198810,"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-eddcc8ed7e6906a2"]},{"id":198811,"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-eddcc8ed7e6906a2"]},{"id":198812,"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-eddcc8ed7e6906a2"]}],"region_exact":{},"incoming_edges":[{"id":"e-eddcc8ed7e6906a2","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-bc12291099a8ab59","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":"727476ca235e54313ec34fc973d320834d8356eba9e5e8cfa58d7416109be6d9","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-de4284a2d5d7d0a6","role":"sequence","name":"A Motzkin related 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:   1;\n  1:   1,     2;\n  2:   2,     6,     4;\n  3:   4,    18,    24,      8;\n  4:   9,    52,   104,     80,     16;\n  5:  21,   150,   400,    480,    240,     32;\n  6:  51,   432,  1440,   2360,   1920,    672,     64;\n  7: 127,  1246,  4984,  10360,  11760,   6944,   1792,   128;\n  8: 323,  3600, 16800,  42336,  61824,  51968,  23296,  4608,   256;\n  9: 835, 10422, 55584, 164640, 294336, 322560, 209664, 73728, 11520, 512;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (3*n-3+(4*n-4)*x-(4*n-4)*x^2)*P(n-2,x) + (2*n+1+(4*n+2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (4*n - 4)*x - (4*n - 4)*x^2)*P[n - 2, x] + (2*n + 1 + (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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (4*n - 4)*x - (4*n - 4)*x^2)*v[i - 2] + (2*n + 1 + (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-de4284a2d5d7d0a6","steps":["basis A013609"],"edges":["e-eddcc8ed7e6906a2"],"text":"A107131 \u2192 basis A013609"},"headline":"A107131 \u2192 basis A013609","chain":null}