{"id":"n-66cdc186e4602e6f","revision":0,"first_index":1,"fingerprint":"f6-f4880ddb0267db87","name":"basis_map_A013611 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","4"],["3","16","16"],["11","72","128","64"],["49","372","896","832","256"],["261","2212","6480","8384","4864","1024"],["1631","15040","50624","81280","66560","26624","4096"],["11743","115784","432752","810816","833280","474112","139264","16384"],["95901","998788","4054160","8540672","10343424","7440384","3125248","704512","65536"],["876809","9560996","41507488","95947648","131763968","112458752","60039168","19431424","3473408","262144"],["8877691","100687888","462393072","1153128704","1750446848","1704579072","1085263872","448266240","115408896","16777216","1048576"],["98641011","1157716456","5577901024","14823587904","24432001280","26444607488","19301130240","9545367552","3146711040","660602880","79691776","4194304"],["1193556233","14437999092","72519648416","203486959808","359405899776","424462111744","345840484352","196811636736","77934624768","21006909440","3667918848","373293056","16777216"],["15624736141","194171331204","1011670162928","2975836564032","5576878422016","7089202481152","6327302795264","4033725808640","1843717472256","598568599552","134501892096","19851640832","1728053248","67108864"],["220048367319","2801812923520","15081501987744","46242385594496","91243651789056","123552005564416","119116266041344","83453358669824","42855449821184","16101315969024","4367787753472","831302664192","105126035456","7918845952","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","35","275","2405","23225","245855","2834135","35368625","475321805","6846878075","105272152955","1721157819485","29822847068225","545980200462455"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3"],["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","4"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":205174,"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-1b068b4c605e1c6e"]},{"id":205175,"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-1b068b4c605e1c6e"]},{"id":205176,"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-1b068b4c605e1c6e"]},{"id":205177,"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-1b068b4c605e1c6e"]}],"region_exact":{},"incoming_edges":[{"id":"e-1b068b4c605e1c6e","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-66cdc186e4602e6f","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":"003d70e891aa5d9330b3e1f57876b69c0f431aac0d420bb6ade794cdb27f479c","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,       4;\n   3:      3,      16,       16;\n   4:     11,      72,      128,       64;\n   5:     49,     372,      896,      832,       256;\n   6:    261,    2212,     6480,     8384,      4864,      1024;\n   7:   1631,   15040,    50624,    81280,     66560,     26624,     4096;\n   8:  11743,  115784,   432752,   810816,    833280,    474112,   139264,    16384;\n   9:  95901,  998788,  4054160,  8540672,  10343424,   7440384,  3125248,   704512,   65536;\n  10: 876809, 9560996, 41507488, 95947648, 131763968, 112458752, 60039168, 19431424, 3473408, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n-3)*P(n-2,x) + (n+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n - 3)*P[n - 2, x] + (n + 4*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 + 4*x; for(n = 3, N, my(i = n); v[i] = -(n - 3)*v[i - 2] + (n + 4*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 A013611"],"edges":["e-1b068b4c605e1c6e"],"text":"A269951 \u2192 basis A013611"},"headline":"A269951 \u2192 basis A013611","chain":null}