{"id":"n-bb32e2bdb3184c38","revision":1,"first_index":0,"fingerprint":"f6-3a96f023c411efeb","name":"basis_map_A013610 of Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1"],["7","18"],["25","72"],["61","180"],["481","2520","3240"],["2731","15750","22680"],["10417","61488","90720"],["91225","727272","1905120","1632960"],["681121","5898960","17010000","16329600"],["3493711","30938490","91309680","89812800"],["33597961","361251000","1439998560","2514758400","1616630400"],["303321877","3506410908","15183923040","29189160000","21016195200"],["1938378625","22890794472","101354493240","199420341120","147113366400"]],"degrees":[0,0,0,1,1,1,2,2,2,3,3,3,4,4,4],"row_sums":["1","1","1","25","97","241","6241","41161","162625","4356577","39919681","215554681","5966236321","69199011025","472717373857"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","18"],["-9","-27"],["3","9"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167241,"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-a4ef1ca6ba76aac8"]},{"id":167242,"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-a4ef1ca6ba76aac8"]},{"id":167243,"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-a4ef1ca6ba76aac8"]},{"id":167244,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-a4ef1ca6ba76aac8"]}],"region_exact":{},"incoming_edges":[{"id":"e-a4ef1ca6ba76aac8","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-bb32e2bdb3184c38","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":"a4931be6ef4de2af0ca9bddb9e2f78a14fc50d83da243c055987b30ac3609502","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b12c32f9a9e37f0c","role":"sequence","name":"Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.","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:     1;\n  3:     7,     18;\n  4:    25,     72;\n  5:    61,    180;\n  6:   481,   2520,    3240;\n  7:  2731,  15750,   22680;\n  8: 10417,  61488,   90720;\n  9: 91225, 727272, 1905120, 1632960;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n^2-9*n+6+(9*n^2-27*n+18)*x)*P(n-3,x) + P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n^2 - 9*n + 6 + (9*n^2 - 27*n + 18)*x)*P[n - 3, 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; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (3*n^2 - 9*n + 6 + (9*n^2 - 27*n + 18)*x)*v[i - 3] + 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-b12c32f9a9e37f0c","steps":["basis A013610"],"edges":["e-a4ef1ca6ba76aac8"],"text":"A118394 \u2192 basis A013610"},"headline":"A118394 \u2192 basis A013610","chain":null}
