{"id":"n-584607888f6a6ba8","revision":0,"first_index":1,"fingerprint":"f6-4a6fdd8c1c3f626a","name":"reversal of basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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"],["4","2"],["16","32","8"],["64","288","192","32"],["256","2048","2560","1024","128"],["1024","12800","25600","17920","5120","512"],["4096","73728","215040","229376","110592","24576","2048"],["16384","401408","1605632","2408448","1720320","630784","114688","8192"],["65536","2097152","11010048","22020096","21626880","11534336","3407872","524288","32768"],["262144","10616832","70778880","181665792","233570304","168689664","71565312","17694720","2359296","131072"],["1048576","52428800","432537600","1384120320","2249195520","2099249152","1192755200","419430400","89128960","10485760","524288"],["4194304","253755392","2537553920","9896460288","19792920576","23091740672","16793993216","7843348480","2353004544","438304768","46137344","2097152"],["16777216","1207959552","14394851328","67175972864","161942077440","230317621248","207634825216","123211874304","48771366912","12750684160","2113929216","201326592","8388608"],["67108864","5670699008","79389786112","436643823616","1247553781760","2120841428992","2313645195264","1690740719616","845370359808","290078064640","67175972864","10032775168","872415232","33554432"],["268435456","26306674688","427483463680","2735894167552","9135932309504","18271864619008","23670370074624","20809116549120","12745583886336","5498095009792","1663897174016","345744867328","46976204800","3758096384","134217728"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","56","576","6016","62976","659456","6905856","72318976","757334016","7930904576","83053510656","869747654656","9108115685376","95381425750016"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["4","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-8","-4"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185641,"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-d57ffd70b1fe5af1"]},{"id":185642,"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-d57ffd70b1fe5af1"]},{"id":185643,"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-d57ffd70b1fe5af1"]}],"region_exact":{},"incoming_edges":[{"id":"e-d57ffd70b1fe5af1","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-584607888f6a6ba8","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"8b2c80b2842b43b4b87f7059e907563c3be4c21cbf33d8890a2ce1a82eb7211e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7ff58d52d83b38f4","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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":[{"id":"l-001f048e1b368754","kind":"fingerprint_related","node_a_id":"n-2496e2f555756362","node_b_id":"n-584607888f6a6ba8","fingerprint":"f6-4a6fdd8c1c3f626a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4a6fdd8c1c3f626a"},"other_id":"n-2496e2f555756362","other":{"id":"n-2496e2f555756362","name":"Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","first_index":0,"fingerprint":"f6-4a6fdd8c1c3f626a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","4","2"]],"card_rows":[["1"],["1","2"],["1","4","2"],["1","6","9","2"]],"row_sums_preview":["1","3","7","18","47","123","322","843"],"regions":["(-inf, +inf)","(-inf, 0)"],"region_exact":{},"oeis_ids":["A084534"],"derivation":{"root":"n-2496e2f555756362","steps":[],"edges":[],"text":"A084534"},"headline":"A084534: Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","chain":null}},{"id":"l-131a0372bb15918a","kind":"fingerprint_related","node_a_id":"n-584607888f6a6ba8","node_b_id":"n-8174fc58d16e6b93","fingerprint":"f6-4a6fdd8c1c3f626a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4a6fdd8c1c3f626a"},"other_id":"n-8174fc58d16e6b93","other":{"id":"n-8174fc58d16e6b93","name":"basis_map_A013611 of veronese_section_2_0 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-4a6fdd8c1c3f626a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["8","32","16"]],"card_rows":[["1"],["2","4"],["8","32","16"],["32","192","288","64"]],"row_sums_preview":["1","6","56","576","6016","62976","659456","6905856"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["Veronese 2","basis A013611"],"edges":["e-f72875fb5d524bf0","e-60831ed92a5cea7b"],"text":"A034870 \u2192 Veronese 2 \u2192 basis A013611"},"headline":"A034870 \u2192 Veronese 2 \u2192 basis A013611","chain":null}},{"id":"l-6b0a7d769b2c8227","kind":"fingerprint_related","node_a_id":"n-584607888f6a6ba8","node_b_id":"n-79be028962f4a4ca","fingerprint":"f6-4a6fdd8c1c3f626a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4a6fdd8c1c3f626a"},"other_id":"n-79be028962f4a4ca","other":{"id":"n-79be028962f4a4ca","name":"basis_map_A013611 of hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).","first_index":1,"fingerprint":"f6-4a6fdd8c1c3f626a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["8","32","16"],["32","192","288","64"],["128","1024","2560","2048","256"]],"card_rows":[["8","32","16"],["32","192","288","64"],["128","1024","2560","2048","256"],["512","5120","17920","25600","12800","1024"]],"row_sums_preview":["56","576","6016","62976","659456","6905856","72318976","757334016"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dbc23e13bb290987","steps":["basis A013611"],"edges":["e-b9b8c23af0e7fc0b"],"text":"Hadamard-factor parent of A155495 \u2192 basis A013611"},"headline":"Hadamard-factor parent of A155495 \u2192 basis A013611","chain":null}},{"id":"l-c8180ecc11d7ba6a","kind":"fingerprint_related","node_a_id":"n-3a859c49ad305108","node_b_id":"n-584607888f6a6ba8","fingerprint":"f6-4a6fdd8c1c3f626a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4a6fdd8c1c3f626a"},"other_id":"n-3a859c49ad305108","other":{"id":"n-3a859c49ad305108","name":"reversal of Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","first_index":0,"fingerprint":"f6-4a6fdd8c1c3f626a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["2","4","1"]],"card_rows":[["1"],["2","1"],["2","4","1"],["2","9","6","1"]],"row_sums_preview":["1","3","7","18","47","123","322","843"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2496e2f555756362","steps":["reverse"],"edges":["e-ad5d9d34c87449ad"],"text":"A084534 \u2192 reverse"},"headline":"A084534 \u2192 reverse","chain":null}},{"id":"l-f5555dd0eaf8635d","kind":"fingerprint_related","node_a_id":"n-7ff58d52d83b38f4","node_b_id":"n-584607888f6a6ba8","fingerprint":"f6-4a6fdd8c1c3f626a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4a6fdd8c1c3f626a"},"other_id":"n-7ff58d52d83b38f4","other":{"id":"n-7ff58d52d83b38f4","name":"basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","first_index":1,"fingerprint":"f6-4a6fdd8c1c3f626a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["8","32","16"]],"card_rows":[["1"],["2","4"],["8","32","16"],["32","192","288","64"]],"row_sums_preview":["1","6","56","576","6016","62976","659456","6905856"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A013611"],"edges":["e-11143fa0a471b62a"],"text":"A086645 \u2192 basis A013611"},"headline":"A086645 \u2192 basis A013611","chain":null}}],"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:      4,        2;\n   3:     16,       32,        8;\n   4:     64,      288,      192,        32;\n   5:    256,     2048,     2560,      1024,       128;\n   6:   1024,    12800,    25600,     17920,      5120,       512;\n   7:   4096,    73728,   215040,    229376,    110592,     24576,     2048;\n   8:  16384,   401408,  1605632,   2408448,   1720320,    630784,   114688,     8192;\n   9:  65536,  2097152, 11010048,  22020096,  21626880,  11534336,  3407872,   524288,   32768;\n  10: 262144, 10616832, 70778880, 181665792, 233570304, 168689664, 71565312, 17694720, 2359296, 131072;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (4*n-8+(2*n-4)*x)*P(n-1,x) + (8*x+2*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (2*n - 4)*x)*P[n - 1, x] + (8*x + 2*x^2)*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = ((4*n - 8 + (2*n - 4)*x)*v[i - 1] + (8*x + 2*x^2)*deriv(v[i - 1], x))/(n - 2)); 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-f7935c36a3c0ec8c","steps":["basis A013611","reverse"],"edges":["e-11143fa0a471b62a","e-d57ffd70b1fe5af1"],"text":"A086645 \u2192 basis A013611 \u2192 reverse"},"headline":"A086645 \u2192 basis A013611 \u2192 reverse","chain":null}
