{"id":"n-9796bf969c11399f","revision":0,"first_index":1,"fingerprint":"f6-5da2aa11242d6d62","name":"basis_map_A013610 of Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","3"],["3","12","9"],["11","54","72","27"],["49","279","504","351","81"],["261","1659","3645","3537","1539","243"],["1631","11280","28476","34290","21060","6318","729"],["11743","86838","243423","342063","263655","112509","24786","2187"],["95901","749091","2280465","3603096","3272724","1765638","556227","94041","6561"],["876809","7170747","23347962","40477914","41690943","26686989","10685682","2593782","347733","19683"],["8877691","75515916","260096103","486476172","553852323","404504604","193153653","59836320","11553921","1259712","59049"],["98641011","868287342","3137569326","6253701147","7730437905","6275429316","3435186510","1274152761","315026415","49601160","4487724","177147"],["1193556233","10828499319","40792302234","85846061169","113718272976","100726848783","61552176048","26271182223","7802262468","1577297205","206553402","15766083","531441"],["15624736141","145628498403","569064466647","1255431050451","1764559188216","1682300979414","1126123959411","538437398895","184579930656","44943335514","7574274279","838436751","54738423","1594323"],["220048367319","2101359692640","8483344868106","19508506422678","28870061698881","29319470070462","21200136216831","11139678674982","4290384006909","1208962258218","245965479903","35110181106","3330009306","188130114","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","24","164","1264","10884","103784","1087204","12423744","153898244","2055185464","29442697764","450531309584","7335162587524","126626550860424"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3"],["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","3"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":202349,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-29e09f921a50b8b9"]},{"id":202350,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-29e09f921a50b8b9"]},{"id":202351,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-29e09f921a50b8b9"]},{"id":202352,"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-29e09f921a50b8b9"]}],"region_exact":{},"incoming_edges":[{"id":"e-29e09f921a50b8b9","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-9796bf969c11399f","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":"1e6ca72ad4222927a68bc83b60ee15b1997e24740db2b124b138e9449043e27f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-cd3918419c5686a8","role":"sequence","name":"Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      1,       3;\n   3:      3,      12,        9;\n   4:     11,      54,       72,       27;\n   5:     49,     279,      504,      351,       81;\n   6:    261,    1659,     3645,     3537,     1539,      243;\n   7:   1631,   11280,    28476,    34290,    21060,     6318,      729;\n   8:  11743,   86838,   243423,   342063,   263655,   112509,    24786,    2187;\n   9:  95901,  749091,  2280465,  3603096,  3272724,  1765638,   556227,   94041,   6561;\n  10: 876809, 7170747, 23347962, 40477914, 41690943, 26686989, 10685682, 2593782, 347733, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n-3)*P(n-2,x) + (n+3*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(n - 3)*P[n - 2, x] + (n + 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 3*x; for(n = 3, N, my(i = n); v[i] = -(n - 3)*v[i - 2] + (n + 3*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cd3918419c5686a8","steps":["basis A013610"],"edges":["e-29e09f921a50b8b9"],"text":"A269951 \u2192 basis A013610"},"headline":"A269951 \u2192 basis A013610","chain":null}
