{"id":"n-4165ce68a3ab41a3","revision":0,"first_index":1,"fingerprint":"f6-b24a5168263ea116","name":"theta_plus_one 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"],["2","8"],["8","64","48"],["32","384","864","256"],["128","2048","7680","8192","1280"],["512","10240","53760","102400","64000","6144"],["2048","49152","331776","917504","1075200","442368","28672"],["8192","229376","1892352","6881280","12042240","9633792","2809856","131072"],["32768","1048576","10223616","46137344","108134400","132120576","77070336","16777216","589824"],["131072","4718592","53084160","286261248","843448320","1401421824","1271660544","566231040","95551488","2621440"],["524288","20971520","267386880","1677721600","5963776000","12595494912","15744368640","11072962560","3892838400","524288000","11534336"],["2097152","92274688","1314914304","9412018176","39216742400","100763959296","161642184704","158343364608","89068142592","25375539200","2791309312","50331648"],["8388608","402653184","6341787648","51002736640","243856834560","739271245824","1453443776512","1842540969984","1457478696960","671759728640","158343364608","14495514624","218103808"],["33554432","1744830464","30098325504","268703891456","1450390323200","5072222158848","11835185037312","18509161562112","19087572860928","12475537817600","4803082059776","952677433344","73719087104","939524096"],["134217728","7516192768","140928614400","1382979469312","8319485870080","32988570058752","89219087204352","166472932392960","213033330671616","182718646190080","100495255404544","32830730010624","5557285027840","368293445632","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","120","1536","19328","237056","2846720","33628160","392134656","4525129728","51771867136","588072878080","6639163801600","74561068466176","833539201302528"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","8"]],"recurrence":{"denominator":{"coeffs":[["6","-12"],["-7","26"],["2","-18"],["0","4"]]},"terms":[{"coeff":{"coeffs":[["24","96","32"],["-28","-64","96"],["8","-24","-152"],["0","16","48"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","28","16"],["0","-4","-8","32"],["0","0","-8","-32"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185628,"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-b829a67227f30a94"]},{"id":185629,"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-b829a67227f30a94"]},{"id":185630,"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-b829a67227f30a94"]},{"id":185631,"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-b829a67227f30a94"]}],"region_exact":{},"incoming_edges":[{"id":"e-b829a67227f30a94","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-4165ce68a3ab41a3","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"513e48178e757228c9a459204909c5563834801cada6c753ddf98affd46d9438","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-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-4cf51e5662a7b5bb","kind":"fingerprint_related","node_a_id":"n-4fb262f896308a6d","node_b_id":"n-4165ce68a3ab41a3","fingerprint":"f6-b24a5168263ea116","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b24a5168263ea116"},"other_id":"n-4fb262f896308a6d","other":{"id":"n-4fb262f896308a6d","name":"theta_plus_one 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-b24a5168263ea116","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","8","6"]],"card_rows":[["1"],["1","4"],["1","8","6"],["1","12","27","8"]],"row_sums_preview":["1","5","15","48","151","463","1390","4105"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2496e2f555756362","steps":["\u03b8+1"],"edges":["e-ce62b41b86f6d6ff"],"text":"A084534 \u2192 \u03b8+1"},"headline":"A084534 \u2192 \u03b8+1","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:      2,       8;\n   3:      8,      64,       48;\n   4:     32,     384,      864,       256;\n   5:    128,    2048,     7680,      8192,      1280;\n   6:    512,   10240,    53760,    102400,     64000,       6144;\n   7:   2048,   49152,   331776,    917504,   1075200,     442368,      28672;\n   8:   8192,  229376,  1892352,   6881280,  12042240,    9633792,    2809856,    131072;\n   9:  32768, 1048576, 10223616,  46137344, 108134400,  132120576,   77070336,  16777216,   589824;\n  10: 131072, 4718592, 53084160, 286261248, 843448320, 1401421824, 1271660544, 566231040, 95551488, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-7*n+6+(4*n^3-18*n^2+26*n-12)*x)*P(n,x) = (8*n^2-28*n+24+(16*n^3-24*n^2-64*n+96)*x+(48*n^3-152*n^2+96*n+32)*x^2)*P(n-1,x) - ((4*n-6)*x+(8*n^2+8*n-28)*x^2+(32*n^2-32*n-16)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 8*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 28*n + 24 + (16*n^3 - 24*n^2 - 64*n + 96)*x + (48*n^3 - 152*n^2 + 96*n + 32)*x^2)*P[n - 1, x] - ((4*n - 6)*x + (8*n^2 + 8*n - 28)*x^2 + (32*n^2 - 32*n - 16)*x^3)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6 + (4*n^3 - 18*n^2 + 26*n - 12)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 8*x; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 28*n + 24 + (16*n^3 - 24*n^2 - 64*n + 96)*x + (48*n^3 - 152*n^2 + 96*n + 32)*x^2)*v[i - 1] - ((4*n - 6)*x + (8*n^2 + 8*n - 28)*x^2 + (32*n^2 - 32*n - 16)*x^3)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6 + (4*n^3 - 18*n^2 + 26*n - 12)*x)); 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","\u03b8+1"],"edges":["e-11143fa0a471b62a","e-b829a67227f30a94"],"text":"A086645 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A086645 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}