{"id":"n-3226d385218bf3b4","revision":0,"first_index":0,"fingerprint":"f6-1cfe5e80b35ba55f","name":"basis_map_A008277 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","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"],["32","20"],["1024","1680","400"],["32768","107840","62400","8000"],["1048576","6263040","6649600","1984000","160000"],["33554432","346649600","604032000","315520000","57600000","3200000"],["1073741824","18696867840","50423296000","41108480000","12761600000","1574400000","64000000"],["34359738368","993711964160","4004414668800","4790446080000","2251468800000","463052800000","41216000000","1280000000"],["1099511627776","52360216903680","308192095436800","520809332736000","347973427200000","106152345600000","15525888000000","1044480000000","25600000000"],["35184372088832","2744721511546880","23237035209523200","54078300520448000","49389210501120000","20971578163200000","4482981888000000","490168320000000","25804800000000","512000000000"],["1125899906842624","143429206042214400","1727960965316608000","5439944352071680000","6613157586534400000","3756032527564800000","1100844810240000000","173968588800000000","14757888000000000","624640000000000","10240000000000"],["36028797018963968","7480836712332001280","127281773623640064000","535034099696926720000","849472536733286400000","628059445369241600000","242449061707776000000","51939493478400000000","6312886272000000000","427581440000000000","14868480000000000","204800000000000"],["1152921504606846976","389724084981643345920","9313904435148724633600","51768772644590059520000","105841606108066611200000","99893297523405619200000","49413446286966784000000","13782574112440320000000","2250864033792000000000","216904990720000000000","12001116160000000000","348979200000000000","4096000000000000"],["36893488147419103232","20288710849137590927360","678395601030341040537600","4949005172005259968512000","12889635336995261644800000","15302747395250873958400000","9508709786087063552000000","3358871673079070720000000","707817376736870400000000","91001138708480000000000","7122358763520000000000","327965081600000000000","8093696000000000000","81920000000000000"],["1180591620717411303424","1055750833918103110287360","49250257491167306737254400","468876387845090737913856000","1542619261183574503587840000","2277755362913020595404800000","1751378835390251139072000000","767900123491341434880000000","203078369795060531200000000","33448588940935168000000000","3472410007306240000000000","225094375833600000000000","8760786944000000000000","185794560000000000000","1638400000000000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","52","3104","211008","16105216","1360556032","125702385664","12579950051328","1353182897504256","155455497285926912","18971856704476545024","2448508817110621749248","332884601332976368746496","47513967583691626294280192","7099070571079024077317341184"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["32","20"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","20"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197980,"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-afd4691742acaf5a"]},{"id":197981,"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-afd4691742acaf5a"]},{"id":197982,"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-afd4691742acaf5a"]}],"region_exact":{},"incoming_edges":[{"id":"e-afd4691742acaf5a","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-3226d385218bf3b4","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"38fdfe71d608e5c0167e7752a490105d3ad2623c0124beaa677c71afee4e7303","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-384c17a53af3442c","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","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":[{"id":"l-a00e9b1548d42499","kind":"fingerprint_related","node_a_id":"n-034190847bef139d","node_b_id":"n-3226d385218bf3b4","fingerprint":"f6-1cfe5e80b35ba55f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1cfe5e80b35ba55f"},"other_id":"n-034190847bef139d","other":{"id":"n-034190847bef139d","name":"basis_map_A008277 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-1cfe5e80b35ba55f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["8","5"],["64","105","25"]],"card_rows":[["1"],["8","5"],["64","105","25"],["512","1685","975","125"]],"row_sums_preview":["1","13","194","3297","62911","1328668","30689059","767819217"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A008277"],"edges":["e-f7e2151b905e00c8"],"text":"A013622 \u2192 basis A008277"},"headline":"A013622 \u2192 basis A008277","chain":null}},{"id":"l-c0c7552f4e58bbb7","kind":"fingerprint_related","node_a_id":"n-f8e5ffc520f5b05b","node_b_id":"n-3226d385218bf3b4","fingerprint":"f6-1cfe5e80b35ba55f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1cfe5e80b35ba55f"},"other_id":"n-f8e5ffc520f5b05b","other":{"id":"n-f8e5ffc520f5b05b","name":"basis_map_A008277 of basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-1cfe5e80b35ba55f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["256","420","100"],["4096","13480","7800","1000"],["65536","391440","415600","124000","10000"]],"card_rows":[["256","420","100"],["4096","13480","7800","1000"],["65536","391440","415600","124000","10000"],["1048576","10832800","18876000","9860000","1800000","100000"]],"row_sums_preview":["776","26376","1006576","42517376","1964099776","98280859776","5285870693376","303624018136576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013609","basis A008277"],"edges":["e-31086658f6c15e80","e-bbddc24cbe0ad2ab"],"text":"A013612 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A013612 \u2192 basis A013609 \u2192 basis A008277","chain":null}}],"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:             32,               20;\n  2:           1024,             1680,               400;\n  3:          32768,           107840,             62400,              8000;\n  4:        1048576,          6263040,           6649600,           1984000,            160000;\n  5:       33554432,        346649600,         604032000,         315520000,          57600000,           3200000;\n  6:     1073741824,      18696867840,       50423296000,       41108480000,       12761600000,        1574400000,         64000000;\n  7:    34359738368,     993711964160,     4004414668800,     4790446080000,     2251468800000,      463052800000,      41216000000,      1280000000;\n  8:  1099511627776,   52360216903680,   308192095436800,   520809332736000,   347973427200000,   106152345600000,   15525888000000,   1044480000000,    25600000000;\n  9: 35184372088832, 2744721511546880, 23237035209523200, 54078300520448000, 49389210501120000, 20971578163200000, 4482981888000000, 490168320000000, 25804800000000, 512000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (32+20*x)*P(n-1,x) + 20*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(32 + 20*x)*P[n - 1, x] + 20*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (32 + 20*x)*v[i - 1] + 20*x*deriv(v[i - 1], x)); 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-a6a633b989da1879","steps":["basis A013611","basis A008277"],"edges":["e-62a1b0a86f60f435","e-afd4691742acaf5a"],"text":"A038271 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A038271 \u2192 basis A013611 \u2192 basis A008277","chain":null}